< 0$: opens downwards (maximum)","points":2},{"id":"0f5df822-ed2d-4501-8df7-4a8768828982","cardType":"cloze","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 $\\boxed{\\Delta = b^2 - 4ac}$.","points":null},{"id":"6c727091-aa90-4870-aa6f-092ab46c7b3d","cardType":"basic","front":"How many real roots exist if the discriminant $\\Delta > 0$?","back":"Two distinct real roots.","points":1},{"id":"545b8411-f416-49b5-90fb-8655eca284e0","cardType":"cloze","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 $\\boxed{\\Delta = 0}$.","points":null},{"id":"89c2b5d8-c7cc-48aa-8913-a630acb1a034","cardType":"basic","front":"What does $\\Delta < 0$ imply about the graph of a quadratic?","back":"The curve never crosses the $x$-axis (no real roots).","points":1},{"id":"12c91f20-0be5-45b1-8509-f38c3aef8695","cardType":"basic","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.","points":1},{"id":"86d32ffa-50ac-4908-b07d-765be22f9f5e","cardType":"cloze","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.","points":null},{"id":"acf110db-230b-4d60-b189-e3c5ded78da7","cardType":"basic","front":"Identify the turning point of $y = a(x - h)^2 + k$.","back":"$$(h, k)$","points":1},{"id":"6d7565b2-9fb1-453a-ae68-8c090d261bdf","cardType":"cloze","front":"In the form $y = a(x + b)^2 + c$, the turning point is located at $\\text{[...]}$.","back":"In the form $y = a(x + b)^2 + c$, the turning point is located at $\\boxed{(-b, c)}$.","points":null},{"id":"cb578e14-b6e7-4d74-a5d3-532d3caf4cdc","cardType":"basic","front":"What is the equation for the axis of symmetry of $y = a(x - h)^2 + k$?","back":"$$x = h$","points":1},{"id":"338d9ce3-4cac-4bd1-8b89-4bcd3f21f95c","cardType":"basic","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 $> 0$

","points":3},{"id":"0f6c8b6f-7c36-4536-b892-3b8a36e04756","cardType":"basic","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 ($\\Delta = 0$).","points":1},{"id":"48dabfaa-bd9b-42a3-8f2e-87cf2e0e33ed","cardType":"basic","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 $k$

","points":3},{"id":"1f7d0120-8adb-43af-b3e4-2dd328d0a262","cardType":"basic","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).","points":1},{"id":"4972bff8-d466-436e-a6d0-188d591449ea","cardType":"cloze","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 $\\ge 0$.","points":null},{"id":"f9457138-f71c-4bc9-be77-3ddec6bc3697","cardType":"cloze","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 $\\boxed{\\ge 0}$.","p

Quadratics Flashcards

Quadratics Flashcards

Quadratics Flashcards

Quadratics Flashcards