$2$ . $8$ $-112$ d. Factor and use the Zero Product Property to find the solutions of the quadratic equation: $\newline$ $0=x^{2}+12 x+36$

Full solution

2

8

-112

d. Factor and use the Zero Product Property to find the solutions of the quadratic equation:

\newline

0=x^{2}+12 x+36

Identify Coefficients: Identify the coefficients of $a$ , $b$ , and $c$ in the quadratic equation $x^2 + 12x + 36 = 0$ . By comparing $x^2 + 12x + 36$ with the standard quadratic form $ax^2 + bx + c = 0$ , we find that $a = 1$ , $b = 12$ , and $c = 36$ .
Factor the Equation: Factor the quadratic equation. $\newline$ We look for two numbers that multiply to $ac$ ( $1\times36 = 36$ ) and add up to $b$ ( $12$ ). The numbers $6$ and $6$ fit this requirement because $6\times6 = 36$ and $6+6 = 12$ . $\newline$ Therefore, we can factor the quadratic as $(x + 6)(x + 6) = 0$ .
Apply Zero Product Property: Apply the Zero Product Property. $\newline$ The Zero Product Property states that if a product of factors equals zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero: $\newline$ $x + 6 = 0$ $\newline$ $x + 6 = 0$
Solve for x: Solve each equation for x. $\newline$ Subtract $6$ from both sides of the first equation: $\newline$ $x + 6 - 6 = 0 - 6$ $\newline$ $x = -6$ $\newline$ Since both factors are the same, the second equation will give the same solution: $\newline$ $x + 6 - 6 = 0 - 6$ $\newline$ $x = -6$
State the Roots: State the roots of the equation. $\newline$ The roots of the equation $x^2 + 12x + 36 = 0$ are $x = -6$ and $x = -6$ . Since both roots are the same, we have a repeated root, and the solution can be expressed as a single root $x = -6$ .