Quadratic Equations - PSAT Math

1. What is the discriminant for $ax^2+bx+c=0$?

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

2. What is the vertex of $y=a(x-h)^2+k$?

   $(h,k)$. The vertex is the point where the parabola reaches its extreme value.

3. What is the discriminant of $3x^2-6x+1=0$?

   $24$. $\Delta = (-6)^2 - 4(3)(1) = 36 - 12 = 24$.

4. What is the product of the solutions of $ax^2+bx+c=0$ in terms of $a$ and $c$?

   $r_1r_2=\frac{c}{a}$. Another application of Vieta's formulas.

5. What is the factored form of a quadratic with zeros $r_1$ and $r_2$?

   $y=a(x-r_1)(x-r_2)$. Each factor equals zero when $x$ equals a root.

6. What is the vertex $x$-coordinate of $y=ax^2+bx+c$?

   $-\frac{b}{2a}$. The $x$-value where the parabola reaches its maximum or minimum.

7. What condition on $b^2-4ac$ gives no real solutions?

   $b^2-4ac<0$. Negative discriminant means no real square root exists.

8. What condition on $b^2-4ac$ gives exactly two distinct real solutions?

   $b^2-4ac>0$. Positive discriminant means the square root yields two different values.

9. State the quadratic formula for solutions of $ax^2+bx+c=0$.

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

10. What is the $y$-intercept of $y=ax^2+bx+c$?

    $(0,c)$. Found by substituting $x=0$ into the equation.

11. Find the axis of symmetry of $y=2x^2-8x+1$.

    $x=2$. Use $x=-\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.

12. What is the vertex form of a quadratic function?

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

13. What is the direction of opening for $y=ax^2+bx+c$ when $a<0$?

    Opens downward. Negative leading coefficient creates an inverted U-shape.

14. Identify the solutions of $x^2-5x+6=0$.

    $x=2$ and $x=3$. Factor as $(x-2)(x-3)=0$, so $x=2$ or $x=3$.

15. What is the sum of the solutions of $ax^2+bx+c=0$ in terms of $a$ and $b$?

    $r_1+r_2=-\frac{b}{a}$. From Vieta's formulas for quadratic equations.

16. Identify the $y$-intercept of $y=ax^2+bx+c$.

    $(0,c)$. Found by setting $x=0$ in the equation.

17. What condition on $b^2-4ac$ gives exactly one real solution (a double root)?

    $b^2-4ac=0$. Zero discriminant makes $\pm\sqrt{0} = 0$, giving one repeated solution.

18. What does $\Delta<0$ tell you about the solutions of $ax^2+bx+c=0$?

    No real solutions (two complex solutions). Negative under square root means no real solutions exist.

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

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

20. Identify the direction the parabola opens for $y=ax^2+bx+c$ when $a>0$.

    Opens upward. Positive leading coefficient creates a U-shaped parabola.

21. What is the direction of opening for $y=ax^2+bx+c$ when $a>0$?

    Opens upward. Positive leading coefficient creates a U-shaped parabola.

22. Identify the direction the parabola opens for $y=ax^2+bx+c$ when $a<0$.

    Opens downward. Negative leading coefficient creates an inverted U-shape.

23. Solve for $x$: $2x^2+3x-2=0$.

    $x=\frac{1}{2}$ or $x=-2$. Factor as $(2x-1)(x+2)=0$, giving these solutions.

24. What is the axis of symmetry for $y=2x^2-8x+5$?

    $x=2$. Use $x = -\frac{b}{2a} = -\frac{-8}{2(2)} = \frac{8}{4} = 2$.

25. Solve for $x$: $x^2-5x+6=0$.

    $x=2$ or $x=3$. Factor as $(x-2)(x-3)=0$, so $x=2$ or $x=3$.

26. What is the standard form of a quadratic equation in one variable?

    $ax^2+bx+c=0$, where $a\ne 0$. Must have $a \ne 0$ to ensure the equation is quadratic, not linear.

27. What is the factored form of a quadratic with zeros $r_1$ and $r_2$?

    $y=a(x-r_1)(x-r_2)$. Each factor equals zero when $x$ equals a root.

28. What is the $x$-coordinate of the vertex for $y=ax^2+bx+c$?

    $x=-\frac{b}{2a}$. Found by completing the square or using calculus to find the minimum/maximum.

29. What is the vertex form of a quadratic function?

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

30. Identify the solutions of $2x^2+3x-2=0$.

    $x=\frac{1}{2}$ and $x=-2$. Factor as $(2x-1)(x+2)=0$, so $x=\frac{1}{2}$ or $x=-2$.

31. What does $\Delta=0$ tell you about the solutions of $ax^2+bx+c=0$?

    One real solution (a repeated root). Zero discriminant makes $\pm\sqrt{0} = 0$, giving one solution.

32. What is the quadratic formula for solutions to $ax^2+bx+c=0$?

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

33. Find the discriminant of $3x^2-4x+1=0$.

    $4$. Calculate $b^2-4ac = (-4)^2-4(3)(1) = 16-12 = 4$.

34. What does $\Delta>0$ tell you about the solutions of $ax^2+bx+c=0$?

    Two distinct real solutions. Positive discriminant means the square root yields two different values.

35. Find the vertex of $y=(x-3)^2-5$.

    $(3,-5)$. In vertex form $y=a(x-h)^2+k$, the vertex is $(h,k)$.

36. What is the standard form of a quadratic equation in one variable?

    $ax^2+bx+c=0$ with $a\ne 0$. Must have $a \ne 0$ to ensure the equation is quadratic, not linear.

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