Question

what are the solutions for the system of equations? select all that apply.\\( f(x) = x + 10 \\)\\( f(x) = -x^2 + 12 \\)

Explanation:

Step1: Set equations equal

Since $f(x)$ is equal to both expressions, set them equal to each other:
$x + 10 = -x^2 + 12$

Step2: Rearrange to quadratic form

Bring all terms to one side to form a standard quadratic equation:
$x^2 + x + 10 - 12 = 0$
$x^2 + x - 2 = 0$

Step3: Factor the quadratic

Factor the quadratic expression to solve for $x$:
$(x + 2)(x - 1) = 0$

Step4: Solve for x-values

Set each factor equal to zero and solve:
$x + 2 = 0 \implies x = -2$
$x - 1 = 0 \implies x = 1$

Step5: Find corresponding y-values

Substitute $x=-2$ into $f(x)=x+10$:
$f(-2) = -2 + 10 = 8$
Substitute $x=1$ into $f(x)=x+10$:
$f(1) = 1 + 10 = 11$

Answer:

The solutions are the ordered pairs $(-2, 8)$ and $(1, 11)$