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24:[["$","$L33",null,{"feature":"flashcards"}],"$L34",["$","$L35",null,{"href":"/as-level/maths/edexcel/as-pure-mathematics-2-quadratics/flashcards","label":"Quadratics","subjectCode":"as_level_maths","surface":"flashcards"}],["$","$L36",null,{"runner":["$","div",null,{"className":"min-h-0 flex-1 overflow-hidden bg-background","children":["$","$L37",null,{"decks":[{"id":"17fdee83-91cb-4465-be5d-744f7a794173","title":"Quadratics Flashcards","topicId":"b36f0350-3fc3-4cc7-85a1-b08e7a84abca","topicSlug":"as-pure-mathematics-2-quadratics","cards":[{"id":"1d21e90f-b687-43a9-92fa-828483c53b0c","cardType":"basic","points":1,"front":"In the quadratic expression $ax^2 + bx + c$, what is the necessary condition for $a$?","back":"$$a \\neq 0$","mcqOptions":[{"text":"$$a \\neq 0$","correct":true,"explanation":"A quadratic must have a non-zero coefficient of $x^2$. If $a=0$, the squared term disappears and the expression becomes linear or constant."},{"text":"$$a > 0$","correct":false,"explanation":"Requiring $a$ to be positive wrongly excludes downward-opening quadratics. The necessary condition is $a \\neq 0$, since either sign still gives a quadratic."},{"text":"$$a = 0$","correct":false,"explanation":"If $a=0$, the $ax^2$ term vanishes, so the expression is not quadratic. The required condition is therefore $a \\neq 0$."},{"text":"$$a < 0$","correct":false,"explanation":"A negative coefficient is allowed and simply produces a downward-opening parabola. A quadratic only requires $a \\neq 0$, not $a < 0$."}]},{"id":"f5ec254a-078b-42c7-b0ed-7850f925e98f","cardType":"basic","points":2,"front":"What does the sign of $a$ in $y = ax^2 + bx + c$ indicate about the parabola's shape?","back":"

If $a > 0$: opens upwards (minimum)
If $a < 0$: opens downwards (maximum)

","mcqOptions":[{"text":"

If $a > 0$: opens upwards (minimum)
If $a < 0$: opens downwards (maximum)

","correct":true,"explanation":"The leading coefficient controls the parabola’s orientation. A positive value gives an upward-opening curve with a minimum, while a negative value gives a downward-opening curve with a maximum."},{"text":"

If $a > 0$: opens downwards (maximum)
If $a < 0$: opens upwards (minimum)

","correct":false,"explanation":"This reverses the effect of the leading coefficient. Positive $a$ produces an upward-opening parabola with a minimum, while negative $a$ produces a downward-opening parabola with a maximum."},{"text":"

If $a > 0$: opens upwards (minimum)
If $a < 0$: opens upwards (minimum)

","correct":false,"explanation":"The sign of $a$ changes the opening direction, so both cases cannot open upwards. Only $a>0$ opens upwards; $a<0$ opens downwards."},{"text":"

If $a > 0$: opens right (minimum)
If $a < 0$: opens left (maximum)

","correct":false,"explanation":"The sign of $a$ determines whether the parabola opens vertically upwards or downwards, not sideways. Quadratic functions of this form do not open left or right."}]},{"id":"0f5df822-ed2d-4501-8df7-4a8768828982","cardType":"cloze","points":null,"front":"For a quadratic equation $ax^2 + bx + c = 0$, the discriminant $\\Delta$ is given by the formula $\\text{[...]}$.","back":"For a quadratic equation $ax^2 + bx + c = 0$, the discriminant $\\Delta$ is given by the formula $\\boxed{\\Delta = b^2 - 4ac}$.","mcqOptions":[{"text":"$$\\Delta = b^2 - 4ac$","correct":true,"explanation":"For coefficients $a$, $b$, and $c$ in standard quadratic form, the discriminant is obtained by subtracting $4ac$ from $b^2$."},{"text":"$$\\Delta = b^2 + 4ac$","correct":false,"explanation":"Using addition instead of subtraction is a common sign error. The discriminant is $\\Delta=b^2-4ac$, and its sign determines the number of real roots."},{"text":"$$\\Delta = b^2 - 4a^2c^2$","correct":false,"explanation":"This incorrectly squares the entire coefficient product. The discriminant requires only $b$ to be squared, giving $\\Delta=b^2-4ac$."},{"text":"$$\\Delta = (b - 4ac)^2$","correct":false,"explanation":"Squaring $b-4ac$ introduces extra terms and does not give the discriminant. The correct formula is $\\Delta=b^2-4ac$."}]},{"id":"6c727091-aa90-4870-aa6f-092ab46c7b3d","cardType":"basic","points":1,"front":"How many real roots exist if the discriminant $\\Delta > 0$?","back":"Two distinct real roots.","mcqOptions":[{"text":"Two distinct real roots.","correct":true,"explanation":"A positive discriminant makes the quadratic formula’s square-root term non-zero and real, producing two different real values of $x$."},{"text":"One repeated real root.","correct":false,"explanation":"One repeated real root occurs when $\\Delta=0$, not when it is positive. For $\\Delta>0$, the quadratic has two distinct real roots."},{"text":"No real roots.","correct":false,"explanation":"No real roots occur when $\\Delta<0$, because the quadratic formula requires the square root of a negative number. A positive discriminant gives two distinct real roots."},{"text":"Three distinct real roots.","correct":false,"explanation":"A quadratic cannot have three distinct roots because its degree is two. When $\\Delta>0$, it has exactly two distinct real roots."}]},{"id":"545b8411-f416-49b5-90fb-8655eca284e0","cardType":"cloze","points":null,"front":"If a quadratic has a repeated root, its discriminant must be $\\text{[...]}$.","back":"If a quadratic has a repeated root, its discriminant must be $\\boxed{\\Delta = 0}$.","mcqOptions":[{"text":"$$\\Delta = 0$","correct":true,"explanation":"A repeated root occurs when the square-root term in the quadratic formula vanishes. This happens precisely when the discriminant equals zero."},{"text":"$$\\Delta > 0$","correct":false,"explanation":"A positive discriminant produces two distinct real roots rather than a repeated root. Repetition occurs only when $\\Delta=0$."},{"text":"$$\\Delta < 0$","correct":false,"explanation":"A negative discriminant gives no real roots, not a repeated real root. A repeated root requires the boundary case $\\Delta=0$."},{"text":"$$\\Delta = 1$","correct":false,"explanation":"The value one has no special role in identifying a repeated root. The quadratic formula gives equal solutions only when $\\Delta=0$."}]},{"id":"89c2b5d8-c7cc-48aa-8913-a630acb1a034","cardType":"basic","points":1,"front":"What does $\\Delta < 0$ imply about the graph of a quadratic?","back":"The curve never crosses the $x$-axis (no real roots).","mcqOptions":[{"text":"The curve never crosses the $x$-axis (no real roots).","correct":true,"explanation":"A negative discriminant means there are no real solutions to the quadratic equation. Consequently, its graph has no points where $y=0$ and never crosses the $x$-axis."},{"text":"The curve crosses twice the $x$-axis (two real roots).","correct":false,"explanation":"Crossing the $x$-axis twice corresponds to two distinct real roots and therefore $\\Delta>0$. When $\\Delta<0$, the curve never crosses the axis."},{"text":"The curve touches once the $x$-axis (one real root).","correct":false,"explanation":"Touching the $x$-axis once corresponds to a repeated root, for which $\\Delta=0$. A negative discriminant means the curve never meets the axis."},{"text":"The curve crosses once the $x$-axis (one real root).","correct":false,"explanation":"A curve crossing once would require a single simple root, which a genuine quadratic cannot have. For $\\Delta<0$, it never crosses the $x$-axis."}]},{"id":"12c91f20-0be5-45b1-8509-f38c3aef8695","cardType":"basic","points":1,"front":"What is the graphical behavior of a quadratic at its x-intercept when $\\Delta = 0$?","back":"The curve touches the $x$-axis at its turning point.","mcqOptions":[{"text":"The curve touches the $x$-axis at its turning point.","correct":true,"explanation":"When $\\Delta=0$, both solutions coincide, so the repeated root is also the turning point. The curve meets the $x$-axis there without passing through it."},{"text":"The curve crosses the $x$-axis at its turning point.","correct":false,"explanation":"Crossing at the turning point would require the curve to change sides of the axis, which does not happen at a repeated root. It touches the $x$-axis instead."},{"text":"The curve crosses the $x$-axis at two distinct points.","correct":false,"explanation":"Two intersections correspond to two distinct real roots and a positive discriminant. When $\\Delta=0$, the curve touches the $x$-axis once at its turning point."},{"text":"The curve misses the $x$-axis at its turning point.","correct":false,"explanation":"Failing to meet the $x$-axis corresponds to a negative discriminant. When $\\Delta=0$, the curve touches the axis at its turning point."}]},{"id":"86d32ffa-50ac-4908-b07d-765be22f9f5e","cardType":"cloze","points":null,"front":"When solving for a parameter $k$ in $(k+1)x^2 + 2x + k = 0$, you must exclude the value $\\text{[...]}$ because the equation would no longer be quadratic.","back":"When solving for a parameter $k$ in $(k+1)x^2 + 2x + k = 0$, you must exclude the value $\\boxed{k = -1}$ because the equation would no longer be quadratic.","mcqOptions":[{"text":"$$k = -1$","correct":true,"explanation":"Substituting $k=-1$ makes the leading coefficient $k+1$ equal to zero. The $x^2$ term then disappears, leaving a linear equation."},{"text":"$$k = 1$","correct":false,"explanation":"Substituting $k=1$ makes the leading coefficient $k+1=2$, so the equation remains quadratic. The excluded value is $k=-1$, which removes the $x^2$ term."},{"text":"$$k = 0$","correct":false,"explanation":"Setting $k=0$ removes the constant term but leaves the $x^2$ coefficient equal to one, so the equation remains quadratic. The excluded value is $k=-1$."},{"text":"$$k = -2$","correct":false,"explanation":"Substituting $k=-2$ makes the leading coefficient equal to $-1$, which is non-zero, so the equation remains quadratic. Only $k=-1$ must be excluded."}]},{"id":"acf110db-230b-4d60-b189-e3c5ded78da7","cardType":"basic","points":1,"front":"Identify the turning point of $y = a(x - h)^2 + k$.","back":"$$(h, k)$","mcqOptions":[{"text":"$$(h, k)$","correct":true,"explanation":"Vertex form shows that the squared term is zero when $x=h$. Substituting this value gives $y=k$, so the turning point is $(h, k)$."},{"text":"$$(-h, k)$","correct":false,"explanation":"The subtraction inside the bracket already accounts for the horizontal shift: $x-h=0$ when $x=h$, not $x=-h$. The turning point is $(h, k)$."},{"text":"$$(h, -k)$","correct":false,"explanation":"The constant $k$ is the vertical coordinate without a sign reversal. Since the squared term vanishes at $x=h$, the turning point is $(h, k)$."},{"text":"$$(-h, -k)$","correct":false,"explanation":"Both coordinates have been incorrectly sign-reversed. Setting $x-h=0$ gives $x=h$, and the resulting value of $y$ is $k$, so the point is $(h, k)$."}]},{"id":"6d7565b2-9fb1-453a-ae68-8c090d261bdf","cardType":"cloze","points":null,"front":"In the form $y = a(x + b)^2 + c$, the turning point is located at $\\text{[...]}$.","back":"In the form $y = a(x + b)^2 + c$, the turning point is located at $\\boxed{(-b, c)}$.","mcqOptions":[{"text":"$$(-b, c)$","correct":true,"explanation":"The squared term is zero when $x+b=0$, giving $x=-b$. At that input, $y=c$, so the turning point is $(-b, c)$."},{"text":"$$(b, c)$","correct":false,"explanation":"The horizontal translation has the opposite sign to the one inside the bracket. Solving $x+b=0$ gives $x=-b$, so the turning point is $(-b, c)$."},{"text":"$$(-b, -c)$","correct":false,"explanation":"The vertical coordinate is $c$, not $-c$, because the squared term becomes zero at the turning point. Solving $x+b=0$ gives the correct point $(-b, c)$."},{"text":"$$(b, -c)$","correct":false,"explanation":"This reverses both coordinates incorrectly. The bracket vanishes at $x=-b$, and substituting that value leaves $y=c$, so the turning point is $(-b, c)$."}]},{"id":"cb578e14-b6e7-4d74-a5d3-532d3caf4cdc","cardType":"basic","points":1,"front":"What is the equation for the axis of symmetry of $y = a(x - h)^2 + k$?","back":"$$x = h$","mcqOptions":[{"text":"$$x = h$","correct":true,"explanation":"The graph is symmetric about the vertical line through its turning point. Since the turning point has horizontal coordinate $h$, the axis of symmetry is $x=h$."},{"text":"$$x = -h$","correct":false,"explanation":"The sign inside vertex form is often misread. The bracket is zero when $x-h=0$, which gives $x=h$, so the axis is $x=h$, not $x=-h$."},{"text":"$$y = k$","correct":false,"explanation":"Although $y=k$ passes horizontally through the turning point, a quadratic's axis of symmetry is vertical. Its equation is $x=h$, using the turning point's horizontal coordinate."},{"text":"$$x = k$","correct":false,"explanation":"This incorrectly uses the vertical translation as the horizontal coordinate. The turning point is $(h,k)$, so its vertical axis of symmetry has equation $x=h$."}]},{"id":"338d9ce3-4cac-4bd1-8b89-4bcd3f21f95c","cardType":"basic","points":3,"front":"How do you prove that $x^2 + 2x + 9$ is always positive for all real $x$?","back":"

Complete the square: $(x + 1)^2 + 8$
State that $(x + 1)^2 \\ge 0$
Conclude minimum value is $8$, which is $> 0$

","mcqOptions":[{"text":"

Complete the square: $(x + 1)^2 + 8$
State that $(x + 1)^2 \\ge 0$
Conclude minimum value is $8$, which is $> 0$

","correct":true,"explanation":"Completing the square gives $(x+1)^2+8$. Since every real square is at least zero, the expression has minimum value $8$, proving it is always positive."},{"text":"

Complete the square: $(x + 1)^2 + 10$
State that $(x + 1)^2 \\ge 0$
Conclude minimum value is $10$, which is $> 0$

","correct":false,"explanation":"The constant was calculated incorrectly: $(x+1)^2=x^2+2x+1$, so $8$, not $10$, must be added. The correct minimum is therefore $8$."},{"text":"

Complete the square: $(x - 1)^2 + 8$
State that $(x - 1)^2 \\ge 0$
Conclude minimum value is $8$, which is $> 0$

","correct":false,"explanation":"Using $x-1$ produces a middle term of $-2x$, so this completed square represents a different quadratic. The correct form is $(x+1)^2+8$, with minimum $8$."},{"text":"

Complete the square: $(x + 1)^2 - 8$
State that $(x + 1)^2 \\ge 0$
Conclude minimum value is $-8$, which is $< 0$

","correct":false,"explanation":"The constant has the wrong sign: $(x+1)^2-8$ expands to $x^2+2x-7$. The given expression is $(x+1)^2+8$, whose minimum is positive."}]},{"id":"0f6c8b6f-7c36-4536-b892-3b8a36e04756","cardType":"basic","points":1,"front":"What algebraic condition represents a line being tangent to a quadratic curve?","back":"The equation formed by setting them equal has a repeated root ($\\Delta = 0$).","mcqOptions":[{"text":"The equation formed by setting them equal has a repeated root ($\\Delta = 0$).","correct":true,"explanation":"At tangency, the line and curve meet at exactly one repeated solution. Therefore, the quadratic obtained by equating their equations must have discriminant $Delta=b^2-4ac=0$."},{"text":"The equation formed by setting them equal has two distinct roots ($\\Delta > 0$).","correct":false,"explanation":"A positive discriminant gives two distinct intersection points, meaning the line crosses the curve twice. Tangency requires one repeated root, so the correct condition is $\\Delta=0$."},{"text":"The equation formed by setting them equal has no real roots ($\\Delta < 0$).","correct":false,"explanation":"A negative discriminant means there are no real intersection points, so the line does not meet the curve. A tangent produces a repeated root and therefore has $\\Delta=0$."},{"text":"The equation formed by setting them equal has a root at zero ($c = 0$).","correct":false,"explanation":"A root at zero would merely place an intersection on the $y$-axis and does not establish tangency. Tangency requires a repeated root, represented by $\\Delta=0$."}]},{"id":"48dabfaa-bd9b-42a3-8f2e-87cf2e0e33ed","cardType":"basic","points":3,"front":"State the steps to find the values of $k$ for which a line is tangent to a curve.","back":"

Set expressions equal to each other
Rearrange into $ax^2 + bx + c = 0$
Solve $b^2 - 4ac = 0$ for $k$

","mcqOptions":[{"text":"

Set expressions equal to each other
Rearrange into $ax^2 + bx + c = 0$
Solve $b^2 - 4ac = 0$ for $k$

","correct":true,"explanation":"Equating the line and curve finds their intersections. Tangency means the resulting quadratic has one repeated root, so setting its discriminant to zero determines the required values of $k$."},{"text":"

Set expressions equal to each other
Rearrange into $ax^2 + bx + c = 0$
Solve $b^2 - 4ac > 0$ for $k$

","correct":false,"explanation":"Setting the discriminant greater than zero identifies cases with two distinct intersections, not tangency. After equating and rearranging, the required condition is $b^2-4ac=0$."},{"text":"

Add the expressions to each other
Rearrange into $ax^2 + bx + c = 0$
Solve $b^2 - 4ac = 0$ for $k$

","correct":false,"explanation":"Adding the expressions does not produce an equation for their intersection points. They must first be set equal, then rearranged into a quadratic whose discriminant is zero."},{"text":"

Set expressions equal to each other
Rearrange into $ax^2 + bx + c = 0$
Solve $b^2 - 4ac < 0$ for $k$

","correct":false,"explanation":"The condition $b^2-4ac<0$ describes no real intersections, so the line misses the curve. Tangency requires exactly one repeated intersection and hence $b^2-4ac=0$."}]},{"id":"1f7d0120-8adb-43af-b3e4-2dd328d0a262","cardType":"basic","points":1,"front":"In the profit model $P = 50 - 2(x - 12)^2$, what is the maximum profit?","back":"50 (in the model's units, e.g., £50,000).","mcqOptions":[{"text":"50 (in the model's units, e.g., £50,000).","correct":true,"explanation":"Because $(x-12)^2$ is non-negative, the term $-2(x-12)^2$ is at most zero. It equals zero at $x=12$, giving the maximum profit of $50$."},{"text":"12 (in the model's units, e.g., £12,000).","correct":false,"explanation":"The value $12$ is the input at which maximum profit occurs, not the profit itself. Substituting $x=12$ gives $P=50$, so the maximum is $50$."},{"text":"2 (in the model's units, e.g., £2,000).","correct":false,"explanation":"The coefficient $2$ controls how rapidly profit falls away from the maximum; it is not the maximum value. At $x=12$, the squared term vanishes and $P=50$."},{"text":"-50 (in the model's units, e.g., -£50,000).","correct":false,"explanation":"The negative sign applies only to the squared term, not to the constant $50$. Since that term is zero at $x=12$, the maximum profit is positive $50$."}]},{"id":"4972bff8-d466-436e-a6d0-188d591449ea","cardType":"cloze","points":null,"front":"A square of a real expression (e.g., $(x - h)^2$) is always $\\text{[...]}$ or $\\ge 0$.","back":"A square of a real expression (e.g., $(x - h)^2$) is always $\\boxed{non-negative}$ or $\\ge 0$.","mcqOptions":[{"text":"non-negative","correct":true,"explanation":"Squaring any real expression produces either zero or a positive value, never a negative value. Therefore, every real square is described as non-negative."},{"text":"non-positive","correct":false,"explanation":"Non-positive means less than or equal to zero, which reverses the true inequality. A real square is greater than or equal to zero, so it is non-negative."},{"text":"positive","correct":false,"explanation":"A square can equal zero when the expression being squared is zero, so it is not always strictly positive. The precise description is non-negative."},{"text":"negative","correct":false,"explanation":"Squaring removes any negative sign from a real value and cannot produce a negative result. A real square is always greater than or equal to zero, meaning non-negative."}]},{"id":"f9457138-f71c-4bc9-be77-3ddec6bc3697","cardType":"cloze","points":null,"front":"A square of a real expression (e.g., $(x - h)^2$) is always $non-negative$ or $\\text{[...]}$.","back":"A square of a real expression (e.g., $(x - h)^2$) is always $non-negative$ or $\\boxed{\\ge 0}$.","mcqOptions":[{"text":"$$\\ge 0$","correct":true,"explanation":"Squaring any real number produces either zero or a positive value. Therefore, a squared real expression always has a value greater than or equal to zero."},{"text":"$$> 0$","correct":false,"explanation":"A strict inequality wrongly excludes zero. When the expression inside the square equals zero, its square is zero, so the correct condition is $\\ge 0$."},{"text":"$$\\le 0$","correct":false,"explanation":"This reverses the inequality, as squares of non-zero real numbers are positive rather than negative. A real square must satisfy $\\ge 0$."},{"text":"$$= 0$","correct":false,"explanation":"A square equals zero only when the expression being squared is zero. In all other real cases it is positive, so the correct condition is $\\ge 0$."}]},{"id":"8c1739ae-f211-48bf-ab59-0d4aab6caed1","cardType":"basic","points":2,"front":"Why does $(x - 3)^2 + 5 = 0$ have no real roots?","back":"$$(x - 3)^2$ must be $\\ge 0$, so $(x - 3)^2 + 5$ must be $\\ge 5$; it can never equal $0$.","mcqOptions":[{"text":"$$(x - 3)^2$ must be $\\ge 0$, so $(x - 3)^2 + 5$ must be $\\ge 5$; it can never equal $0$.","correct":true,"explanation":"A real square cannot be negative, so its minimum value is $0$. Adding $5$ makes the whole expression at least $5$, preventing it from equalling $0$."},{"text":"$$(x - 3)^2$ must be $\\le 0$, so $(x - 3)^2 + 5$ must be $\\le 5$; it can never equal $0$.","correct":false,"explanation":"The inequality has been reversed: a real square is non-negative, not non-positive. Consequently, the left-hand side is at least $5$, so there are no real roots."},{"text":"$$(x - 3)^2$ must be $\\ge 0$, so $(x - 3)^2 + 5$ must be $\\ge 0$; it can never equal $5$.","correct":false,"explanation":"Adding $5$ raises the minimum from $0$ to $5$, rather than leaving it at $0$. Since the expression is always at least $5$, it cannot equal $0$."},{"text":"$$(x - 3)^2$ can be $-5$, so $(x - 3)^2 + 5$ can be $0$; it has real roots.","correct":false,"explanation":"Real squares cannot take negative values, so the proposed route to zero is impossible. The squared term is at least $0$, making the complete expression at least $5$."}]},{"id":"d439b86f-7548-406e-ae66-fccfc97d72f9","cardType":"basic","points":1,"front":"What is the first step before calculating the discriminant of $3x^2 + 7 = 2x$?","back":"Rearrange to standard form: $3x^2 - 2x + 7 = 0$.","mcqOptions":[{"text":"Rearrange to standard form: $3x^2 - 2x + 7 = 0$.","correct":true,"explanation":"The discriminant formula requires every term on one side and zero on the other. Subtracting $2x$ gives coefficients $a=3$, $b=-2$ and $c=7$."},{"text":"Rearrange to standard form: $3x^2 + 2x + 7 = 0$.","correct":false,"explanation":"Moving $2x$ from the right to the left changes its sign, so it becomes $-2x$, not $+2x$. The correct standard form is $3x^2-2x+7=0$."},{"text":"Rearrange to standard form: $3x^2 - 2x = -7$.","correct":false,"explanation":"Although this equation is equivalent to the original, it is not in the required form $ax^2+bx+c=0$. Moving $-7$ left gives $3x^2-2x+7=0$."},{"text":"Identify the coefficients: $a = 3$, $b = 0$, $c = 7$.","correct":false,"explanation":"Reading coefficients before moving the linear term overlooks the $2x$ on the right. Standard form is $3x^2-2x+7=0$, giving $a=3$, $b=-2$ and $c=7$."}]},{"id":"958465f7-8476-41a6-8c90-1c57e0e98363","cardType":"basic","points":1,"front":"What is the relationship between the turning point and the normal to the curve at that point?","back":"The normal is a vertical line ($x = x_{turning\\,point}$).","mcqOptions":[{"text":"The normal is a vertical line ($x = x_{turning\\,point}$).","correct":true,"explanation":"At a turning point, the tangent to a graph $y=f(x)$ is horizontal and has gradient zero. The perpendicular normal is therefore vertical through the turning point."},{"text":"The normal is a horizontal line ($y = y_{turning\\,point}$).","correct":false,"explanation":"A horizontal line describes the tangent at a stationary turning point, not the normal. The perpendicular normal is the vertical line $x=x_{turning\\,point}$."},{"text":"The tangent is a vertical line ($x = x_{turning\\,point}$).","correct":false,"explanation":"This confuses the tangent with the normal. The tangent is horizontal at the turning point, while the normal is the vertical line $x=x_{turning\\,point}$."},{"text":"The normal has a gradient of zero ($m_{normal} = 0$).","correct":false,"explanation":"Gradient zero belongs to the tangent at a turning point. A line perpendicular to that tangent is vertical, so its gradient is undefined rather than zero."}]},{"id":"89c00219-76c5-41ff-a021-f42256dd69b3","cardType":"cloze","points":null,"front":"To complete the square for $x^2 + px$, you $\\text{[...]}$ the coefficient of $x$ and subtract the $square$ of that new value.","back":"To complete the square for $x^2 + px$, you $\\boxed{halve}$ the coefficient of $x$ and subtract the $square$ of that new value.","mcqOptions":[{"text":"$$halve$","correct":true,"explanation":"Using half of $p$ gives $(x+\\frac{p}{2})^2=x^2+px+\\frac{p^2}{4}$. Subtracting the added square then preserves the original expression."},{"text":"$$double$","correct":false,"explanation":"Doubling the coefficient would produce an inner term whose expansion has an $x$-coefficient of $4p$. The coefficient must instead be halved before forming the bracket."},{"text":"$$square$","correct":false,"explanation":"Squaring the coefficient does not produce the required middle term when the bracket is expanded. The coefficient of $x$ must first be halved."},{"text":"$$negate$","correct":false,"explanation":"Negating the coefficient changes the sign rather than finding the bracket value needed to reproduce $px$. The correct operation is to halve the coefficient."}]},{"id":"8396aff2-9fbf-48fa-b9fa-105d2168a500","cardType":"cloze","points":null,"front":"To complete the square for $x^2 + px$, you $halve$ the coefficient of $x$ and subtract the $\\text{[...]}$ of that new value.","back":"To complete the square for $x^2 + px$, you $halve$ the coefficient of $x$ and subtract the $\\boxed{square}$ of that new value.","mcqOptions":[{"text":"$$square$","correct":true,"explanation":"Expanding $(x+\\frac{p}{2})^2$ introduces the extra constant $(\\frac{p}{2})^2$. Subtracting that square keeps the completed-square expression equivalent to the original."},{"text":"$$cube$","correct":false,"explanation":"Completing the square adds the second power of the halved coefficient, not its third power. The square must be subtracted to cancel the added constant."},{"text":"$$square\\ root$","correct":false,"explanation":"The bracket expansion creates the square of the halved coefficient as its constant term. Subtracting a square root would not cancel that added term."},{"text":"$$reciprocal$","correct":false,"explanation":"Taking a reciprocal is unrelated to the constant generated by expanding the completed square. The added constant is the square of the halved coefficient."}]}]}],"examCountdown":{"examDate":"2027-05-13","paperLabel":"Paper 1"},"heading":"Quadratics","title":"Quadratics Flashcards","topbar":{"backHref":"/as-level/maths/edexcel/flashcards"},"topicSidebar":"$L38"}]}],"topicId":"b36f0350-3fc3-4cc7-85a1-b08e7a84abca","walled":"$L39"}],"$L3a","$L3b"]