Quadratics Core: 17 flashcards
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1. What is the standard form of a quadratic equation?
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The standard form is $ax^2 + bx + c = 0$, where $a \neq 0$, and $a$, $b$, $c$ are real numbers.
2. Why must the coefficient 'a' in a quadratic equation not be zero?
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If $a = 0$, the equation becomes linear ($bx + c = 0$), not quadratic, because the $x^2$ term disappears.
3. What is the quadratic formula used to solve $ax^2 + bx + c = 0$?
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The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
4. How do you solve a quadratic equation by factoring?
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Rewrite the equation in standard form, factor the quadratic expression into two binomials, set each factor equal to zero, and solve for x.
5. What is the zero-product property?
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If the product of two factors is zero, then at least one of the factors must be zero. That is, if $pq = 0$, then $p = 0$ or $q = 0$.
6. Describe the method of completing the square for solving $x^2 + bx = c$.
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Add $(b/2)^2$ to both sides to create a perfect square trinomial on the left, then factor it as $(x + b/2)^2$ and solve by taking the square root.
7. When solving a quadratic equation by taking square roots, what is the key step?
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Isolate the squared term, then take the square root of both sides, remembering to include both the positive and negative roots (e.g., $x^2 = k$ gives $x = \pm \sqrt{k}$).
8. What is the discriminant of a quadratic equation and what does it indicate?
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The discriminant is $D = b^2 - 4ac$. If $D > 0$, there are two distinct real roots; if $D = 0$, there is one real root (repeated); if $D < 0$, there are no real roots (two complex roots).
9. Solve: $x^2 - 5x + 6 = 0$ by factoring.
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Factor to $(x - 2)(x - 3) = 0$, so $x = 2$ or $x = 3$.
10. Solve: $x^2 - 9 = 0$ using square roots.
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Add 9 to both sides to get $x^2 = 9$, then take the square root: $x = \pm 3$.
11. Solve: $2x^2 + 4x - 6 = 0$ using the quadratic formula.
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Here $a=2$, $b=4$, $c=-6$. The discriminant is $4^2 - 4(2)(-6) = 16 + 48 = 64$. Then $x = \frac{-4 \pm 8}{4}$, giving $x = 1$ or $x = -3$.
12. Solve by completing the square: $x^2 + 6x + 5 = 0$.
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Move constant: $x^2 + 6x = -5$. Add $(6/2)^2 = 9$ to both sides: $x^2 + 6x + 9 = 4$, so $(x+3)^2 = 4$. Taking square roots gives $x+3 = \pm 2$, so $x = -1$ or $x = -5$.
13. What is the vertex form of a quadratic equation and how is it related to the standard form?
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Vertex form is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. It is obtained from standard form $y = ax^2 + bx + c$ by completing the square.
14. How do you determine the most efficient method to solve a quadratic equation?
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If the equation can be easily factored, use factoring. If it is in the form $x^2 = k$, use square roots. If the coefficient of $x^2$ is 1 and the coefficient of $x$ is even, completing the square is efficient. Otherwise, use the quadratic formula, which always works.
15. What is the sum and product of the roots of a quadratic equation $ax^2 + bx + c = 0$?
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The sum of the roots is $-b/a$ and the product of the roots is $c/a$.
16. Solve: $3x^2 - 2x - 1 = 0$ using the quadratic formula.
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Here $a=3$, $b=-2$, $c=-1$. Discriminant: $(-2)^2 - 4(3)(-1) = 4 + 12 = 16$. Then $x = \frac{2 \pm 4}{6}$, giving $x = 1$ or $x = -\frac{1}{3}$.
17. What are the roots of the equation $x^2 + 4x + 4 = 0$?
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This is a perfect square trinomial: $(x+2)^2 = 0$, so the root is $x = -2$ (a repeated root).
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