Quadratics - GED Math

1. What does the zero product property state?

   If $ab = 0$, then $a = 0$ or $b = 0$. It is why you set each factor equal to zero.

2. Solve $x^2 + 5x = 0$.

   $x = 0$ or $x = -5$. Factor out $x$: $x(x + 5) = 0$.

3. Solve $x^2 = 49$.

   $x = 7$ or $x = -7$. Both $7^2$ and $(-7)^2$ equal $49$.

4. To complete the square on $x^2 + 8x = 9$, what number do you add to both sides?

   $16$, which is $(\frac{8}{2})^2$. The equation becomes $(x + 4)^2 = 25$.

5. How many real solutions does $x^2 + 2x + 5 = 0$ have?

   None. The discriminant is $2^2 - 4(1)(5) = -16$, which is negative.

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

   $ax^2 + bx + c = 0$, with $a \ne 0$.

7. What must you do to a quadratic equation before reading off $a$, $b$ and $c$ for the quadratic formula?

   Rearrange it into standard form so one side is $0$: $ax^2 + bx + c = 0$.

8. What does the discriminant $b^2 - 4ac$ tell you about a quadratic equation?

   The number of real solutions: two if positive, one if zero, none if negative.

9. Solve $x^2 - 7x + 12 = 0$.

   $x = 3$ or $x = 4$. It factors as $(x - 3)(x - 4) = 0$.

10. What is the quadratic formula?

    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, for an equation in the form $ax^2 + bx + c = 0$.

11. Solve $(x - 2)(x + 5) = 0$.

    $x = 2$ or $x = -5$. Set each factor equal to zero.

12. Factor $x^2 - 16$.

    $(x + 4)(x - 4)$. It is a difference of squares: $a^2 - b^2 = (a + b)(a - b)$.

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