ion $x^2 + bx + c$?","back":"Halve the coefficient of $x$ to get the value inside the bracket: $(x + \\frac{b}{2})^2$.","points":1},{"id":"62ba4c38-4f71-4232-a430-d0e31052fdd1","cardType":"basic","front":"Express $x^2 + 10x + 7$ in completed-square form.","back":"$$(x+5)^2 - 18$","points":2},{"id":"8f11a748-457f-443a-8df7-dae544c58407","cardType":"basic","front":"Why are fractions common when completing the square for $x^2 - 7x + 4$?","back":"Because the coefficient of $x$ ($-7$) is odd.","points":1},{"id":"177adbe7-7f0a-4647-85d2-195b1f128ac2","cardType":"basic","front":"What defines a non-monic quadratic?","back":"The coefficient of $x^2$ is not 1.","points":1},{"id":"9f06e66d-b595-442c-afdf-38b3741ec370","cardType":"basic","front":"What is the first step to complete the square for $3x^2 - 12x + 5$?","back":"Factor out 3 from the $x^2$ and $x$ terms: $3(x^2 - 4x) + 5$.","points":1},{"id":"b0ce053f-48b6-4328-9203-6797c49dc713","cardType":"cloze","front":"When completing the square for $a(x+B)^2+C$, the term $C$ is found by expanding the bracket and adjusting the [ ].","back":"When completing the square for $a(x+B)^2+C$, the term $C$ is found by expanding the bracket and adjusting the constant.","points":null},{"id":"3016fb8e-c7f3-4658-880e-121ed890ae91","cardType":"basic","front":"For a curve $y = (x - a)^2 + b$, what are the coordinates of the minimum point?","back":"$$(a, b)$","points":1},{"id":"136fb390-c9f1-4631-a88f-541ab7288f07","cardType":"basic","front":"State the minimum point of $y = (x + 6)^2 - 11$.","back":"$$(-6, -11)$","points":1},{"id":"ea8ce770-3c9a-4375-9ea1-be35ca23838d","cardType":"cloze","front":"The expression $(x+2)^2$ can never be negative because the result of squaring any real number is always [ ].","back":"The expression $(x+2)^2$ can never be negative because the result of squaring any real number is always $\\ge 0$ (greater than or equal to zero).","points":null},{"id":"5485e393-b334-4e56-9799-9da8b53ba1fb","cardType":"basic","front":"How do you find the $x$-coordinate of the minimum point of $y = (x-4)^2 - 5$?","back":"Set the squared bracket to zero: $x-4=0$, so $x=4$.","points":1},{"id":"19d5002a-7e1f-4d04-a22a-c0683b3f72f5","cardType":"basic","front":"A quadratic has a minimum point at $(3, -4)$. What is its equation in completed-square form?","back":"$$y = (x-3)^2 - 4$","points":1},{"id":"e50c7491-468f-495d-9114-40240afe35ba","cardType":"basic","front":"How can completed-square form prove $x^2 - 10x + 28 > 0$ for all real $x$?","back":"

Rewrite as $(x-5)^2 + 3$.
Since $(x-5)^2 \\ge 0$, the minimum value is 3.
Therefore, the expression is always positive.

","points":3},{"id":"0e3540d1-d870-4628-b265-655fe6e4d1cf","cardType":"basic","front":"What is the first step to prove $x^2 + 3x + 9 > x + 1$ for all real $x$?","back":"Rearrange to one side so the expression is compared to zero: $x^2 + 2x + 8 > 0$.","points":1},{"id":"ea9e5199-f49a-436a-964b-c99190d38575","cardType":"cloze","front":"In the context of quadratic equations, the roots are the $x$-coordinates where the parabola [ ].","back":"In the context of quadratic equations, the roots are the $x$-coordinates where the parabola crosses the $x$-axis.","points":null},{"id":"34bf742d-2e3a-4cf8-8883-d692372c9788","cardType":"basic","front":"Complete the square for $x^2 + px + 5$.","back":"$$(x + \\frac{p}{2})^2 - \\frac{p^2}{4} + 5$","points":2},{"id":"3b9b7470-7c6c-4064-98a4-46295c115888","cardType":"cloze","front":"A quadratic equation has no real roo

2.2 Completing the Square Flashcards

2.2 Completing the Square Flashcards

2.2 Completing the Square Flashcards

2.2 Completing the Square Flashcards