Answer
$x=3$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Express the given equation, $
-x^2=9-6x
,$ in factored form. Then, use the Zero-Factor Property by equating each factor to zero. Finally, solve each equation.
$\bf{\text{Solution Details:}}$
The factored form of the equation above is
\begin{array}{l}\require{cancel}
-x^2+6x-9=0
\\\\
-1(-x^2+6x-9)=0(-1)
\\\\
x^2-6x+9=0
\\\\
(x-3)(x-3)=0
.\end{array}
Equating each factor to zero (Zero-Factor Property), then
\begin{array}{l}\require{cancel}
x-3=0
\text{ OR }
x-3=0
.\end{array}
Using the properties of equality to solve each of the equation above results to
\begin{array}{l}\require{cancel}
x-3=0
\\\\
x=3
\\\\\text{ OR }\\\\
x-3=0
\\\\
x=3
.\end{array}
Hence, the solution is $
x=3
.$
$\bf{\text{Supplementary Solution/s:}}$
To factor the expression, $
x^2-6x+9
,$ find two numbers, $m_1$ and $m_2,$ whose product is $c$ and whose sum is $b$ in the quadratic expression $x^2+bx+c.$ Then, express the factored form as $(x+m_1)(x+m_2).$
$\bf{\text{Solution Details:}}$
In the expression above, the value of $c$ is $
9
$ and the value of $b$ is $
-6
.$
The possible pairs of integers whose product is $c$ are
\begin{array}{l}\require{cancel}
\{ 1,9 \}, \{ 3,3 \},
\{ -1,-9 \}, \{ -3,-3 \}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-3,-3
\}.$ Hence, the factored form of the expression above is
\begin{array}{l}\require{cancel}
(x-3)(x-3)
.\end{array}
