Question

solve the system of equations.
$y = x^2 + 32x - 2$
$y = 20x - 37$
write the coordinates in exact form. simplify all fractions and radicals.
( , )
( , )

Explanation:

Step1: Set the two equations equal

Since both equal \( y \), set \( x^2 + 32x - 2 = 20x - 37 \).

Step2: Rearrange into standard quadratic form

Subtract \( 20x \) and add \( 37 \) to both sides: \( x^2 + 12x + 35 = 0 \).

Step3: Factor the quadratic

Find two numbers that multiply to \( 35 \) and add to \( 12 \): \( (x + 5)(x + 7) = 0 \).

Step4: Solve for \( x \)

Set each factor to zero: \( x + 5 = 0 \) or \( x + 7 = 0 \), so \( x = -5 \) or \( x = -7 \).

Step5: Find corresponding \( y \) values

For \( x = -5 \): \( y = 20(-5) - 37 = -100 - 37 = -137 \).
For \( x = -7 \): \( y = 20(-7) - 37 = -140 - 37 = -177 \).

Answer:

\((-5, -137)\)
\((-7, -177)\)