Question

1. solve the equation by factoring 2x² - 2x = 12
2. solve the equation by factoring x² - 8x + 16 = 0

Explanation:

Step1: Rewrite the first equation in standard form

First, rewrite $2x^{2}-2x = 12$ as $2x^{2}-2x - 12=0$. Then divide through by 2 to simplify: $x^{2}-x - 6 = 0$.

Step2: Factor the simplified first - equation

Factor $x^{2}-x - 6$. We need two numbers that multiply to - 6 and add to - 1. The numbers are - 3 and 2. So, $x^{2}-x - 6=(x - 3)(x + 2)=0$.

Step3: Solve for x in the first - equation

Set each factor equal to zero: $x-3 = 0$ gives $x = 3$; $x + 2=0$ gives $x=-2$.

Step4: Factor the second equation

Factor $x^{2}-8x + 16$. This is a perfect - square trinomial of the form $(a - b)^2=a^{2}-2ab + b^{2}$, where $a = x$ and $b = 4$. So, $x^{2}-8x + 16=(x - 4)^{2}=0$.

Step5: Solve for x in the second - equation

Set $x - 4=0$, then $x = 4$.

Answer:

For the equation $2x^{2}-2x = 12$, the solutions are $x = 3$ and $x=-2$. For the equation $x^{2}-8x + 16 = 0$, the solution is $x = 4$.