Question

solve the system of equations.
y = -5x - 8
y = x² + 12x - 26
write the coordinates in exact form. simplify all fractions and radicals.
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Explanation:

Step1: Set the two equations equal

Since both equations are solved for \( y \), we can set them equal to each other:
\( -5x - 8 = x^2 + 12x - 26 \)

Step2: Rearrange into standard quadratic form

Move all terms to one side to get a quadratic equation:
\( x^2 + 12x + 5x - 26 + 8 = 0 \)
\( x^2 + 17x - 18 = 0 \)

Step3: Factor the quadratic equation

We need two numbers that multiply to \( -18 \) and add to \( 17 \). Those numbers are \( 18 \) and \( -1 \):
\( (x + 18)(x - 1) = 0 \)

Step4: Solve for \( x \)

Set each factor equal to zero:

• \( x + 18 = 0 \) gives \( x = -18 \)
• \( x - 1 = 0 \) gives \( x = 1 \)

Step5: Find the corresponding \( y \)-values

For \( x = -18 \):
Substitute into \( y = -5x - 8 \):
\( y = -5(-18) - 8 = 90 - 8 = 82 \)

For \( x = 1 \):
Substitute into \( y = -5x - 8 \):
\( y = -5(1) - 8 = -5 - 8 = -13 \)

Answer:

\((-18, 82)\)
\((1, -13)\)