Quadratic Equations - SAT Math

1. Find the vertex of $y = -x^2 + 4x - 3$.

   Vertex: $(2, 1)$. Use $x = -\frac{b}{2a} = 2$, then $y = -(4) + 8 - 3 = 1$.

2. Factor $x^2 - 9$.

   $(x - 3)(x + 3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

3. What is the minimum value of $y = x^2 - 6x + 9$?

   Minimum: $0$. This is $(x-3)^2$, so minimum occurs at vertex $(3,0)$.

4. What is the standard form of a quadratic equation?

   $ax^2 + bx + c = 0$. General form where $a \neq 0$ determines parabola shape.

5. Solve for $x$: $x^2 + 2x - 8 = 0$.

   $x = 2, x = -4$. Factor as $(x-2)(x+4) = 0$ or use the quadratic formula.

6. What is the discriminant of $ax^2 + bx + c = 0$?

   $b^2 - 4ac$. The expression under the square root in the quadratic formula.

7. For $y = x^2 + 4x + 4$, what is the vertex?

   Vertex: $(-2, 0)$. This is $(x+2)^2$, so vertex is at $x = -2$, $y = 0$.

8. What is the product of the roots of $ax^2 + bx + c = 0$?

   $\frac{c}{a}$. By Vieta's formulas relating coefficients to roots.

9. Determine the parabola direction for $y = 5x^2 - 3x + 2$.

   Opens upwards. Positive coefficient of $x^2$ means parabola opens upward.

10. How do you determine the number of real roots using the discriminant?

    If $b^2 - 4ac > 0$, 2 real roots; $=0$, 1 real root; $<0$, no real roots. The discriminant determines whether roots are real or complex.

11. Identify the axis of symmetry for $y = ax^2 + bx + c$.

    $x = \frac{-b}{2a}$. Derived from completing the square or using calculus.

12. What is the quadratic formula?

    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Derived from completing the square on the general quadratic equation.

13. What type of parabola does $y = -x^2$ represent?

    A downward-opening parabola. Negative coefficient of $x^2$ makes the parabola open downward.

14. What is the standard form of a quadratic equation?

    $ax^2 + bx + c = 0$. The general form where $a \neq 0$ defines a parabola.

15. What is the vertex of $y = 2(x-1)^2 + 3$?

    Vertex: $(1, 3)$. In vertex form $y = a(x-h)^2 + k$, vertex is $(h,k)$.

16. Find the x-intercepts of $y = x^2 - 4$.

    $x = -2, x = 2$. Set $y = 0$: $x^2 - 4 = 0$, so $x^2 = 4$.

17. State the zero-product property.

    If $ab = 0$, then $a = 0$ or $b = 0$. If a product equals zero, at least one factor must be zero.

18. Convert $y = x^2 - 4x + 4$ to vertex form.

    $y = (x-2)^2$. This is $(x-2)^2$ expanded, so vertex form shows vertex $(2,0)$.

19. Write $x^2 + 8x + 16$ as a square.

    $(x + 4)^2$. Perfect square trinomial with $a = x$, $b = 4$.

20. Find the roots of $x^2 - 5x + 6 = 0$.

    $x = 2, x = 3$. Factor as $(x-2)(x-3) = 0$ to find roots.

21. What is the axis of symmetry for $y = ax^2 + bx + c$?

    $x = \frac{-b}{2a}$. The $x$-coordinate of the vertex, found by calculus or completing the square.

22. What does a positive discriminant indicate about roots?

    Two distinct real roots. When $b^2 - 4ac > 0$, parabola crosses x-axis twice.

23. Convert $y = x^2 + 6x + 8$ to vertex form.

    $y = (x+3)^2 - 1$. Complete the square: $(x+3)^2 = x^2 + 6x + 9$, so subtract 1.

24. What is the effect of increasing $a$ in $y = ax^2$ on the parabola?

    Narrower parabola. Larger $|a|$ values compress the parabola horizontally.

25. What is the vertex form of a quadratic equation?

    $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and vertical shift directly.

26. Identify the $y$-intercept of $y = 2x^2 + 3x + 1$.

    Intercept: $(0, 1)$. Set $x = 0$ to find where the parabola crosses the $y$-axis.

27. State the quadratic formula.

    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Solves any quadratic by substituting coefficients $a$, $b$, and $c$.

28. Find the vertex of $y = x^2 - 4x + 3$.

    Vertex: $(2, -1)$. Use $x = \frac{4}{2} = 2$, then $y = 4 - 8 + 3 = -1$.

29. Find the discriminant for $2x^2 - 4x + 2 = 0$.

    Discriminant: $0$. Calculate $b^2 - 4ac = 16 - 16 = 0$.

30. What is the discriminant in the quadratic formula?

    $b^2 - 4ac$. Expression under the square root that determines root types.

31. Which term in $ax^2 + bx + c$ affects the parabola's direction?

    The coefficient $a$. Positive $a$ opens up, negative $a$ opens down.

32. Simplify: $(x + 2)^2$.

    $x^2 + 4x + 4$. Apply the perfect square formula $(a+b)^2 = a^2 + 2ab + b^2$.

33. Which method can solve any quadratic equation?

    The quadratic formula. Works for any quadratic, unlike factoring or square roots.

34. Identify the vertex form of a quadratic equation.

    $y = a(x-h)^2 + k$. Shows vertex $(h,k)$ and vertical stretch/compression factor $a$.

35. Find the roots of $x^2 - 5x + 6 = 0$.

    $x = 2, x = 3$. Factor as $(x-2)(x-3) = 0$ or use the quadratic formula.

36. State the condition for a perfect square trinomial.

    $b^2 = 4ac$. When discriminant equals zero, the quadratic is a perfect square.

37. What is the sum of the roots of $ax^2 + bx + c = 0$?

    $-\frac{b}{a}$. By Vieta's formulas relating coefficients to roots.

38. Find the axis of symmetry for $y = 3x^2 + 6x + 1$.

    $x = -1$. Use $x = -\frac{b}{2a} = -\frac{6}{2(3)} = -1$.

39. Convert $y = 2(x-3)^2 + 4$ to standard form.

    $y = 2x^2 - 12x + 22$. Expand $(x-3)^2 = x^2 - 6x + 9$, then distribute and combine.

40. Solve $3x^2 - 12 = 0$.

    $x = 2, x = -2$. Divide by 3: $x^2 - 4 = 0$, so $x = \pm 2$.

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