Question

What point(s) are a solution to the equation "y" equals "x" squared plus 4 "x" minus 21y=x2+4x−21 when f(x)= -9? Choose all that apply.

Answer 1

To find the points that are a solution to the equation \( y = x^2 + 4x - 21 \) when \( f(x) = -9 \), we need to set \( y \) equal to \(-9\) and solve for \( x \):

\[
-9 = x^2 + 4x - 21
\]

Now, let's rearrange the equation to set it to zero:

\[
x^2 + 4x - 21 + 9 = 0
\]

This simplifies to:

\[
x^2 + 4x - 12 = 0
\]

Next, we can factor the quadratic equation. We need two numbers that multiply to \(-12\) and add up to \(4\). These numbers are \(6\) and \(-2\). Thus, we can factor the equation as follows:

\[
(x + 6)(x - 2) = 0
\]

Setting each factor equal to zero gives us:

\[
x + 6 = 0 \quad \Rightarrow \quad x = -6
\]
\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]

Now we have \(x\) values \(x = -6\) and \(x = 2\). To find the corresponding \(y\) values, we substitute these \(x\) values back into the equation \(y = -9\):

1. For \(x = -6\),
- \(y = -9\)
- Point: \((-6, -9)\)

2. For \(x = 2\),
- \(y = -9\)
- Point: \((2, -9)\)

Thus, the points that are solutions to the equation when \(f(x) = -9\) are:

- \((-6, -9)\)
- \((2, -9)\)

So, the solutions are the points \((-6, -9)\) and \((2, -9)\).