Question

solve the system of equations.
$y = x^{2}+34x + 24$
$y = 18x - 40$
write the coordinates in exact form. simplify all fractions and radicals.

Explanation:

Step1: Set the two equations equal

Since both equations equal \( y \), we can set them equal to each other:
\( x^2 + 34x + 24 = 18x - 40 \)

Step2: Rearrange into standard quadratic form

Subtract \( 18x \) and add \( 40 \) to both sides:
\( x^2 + 34x - 18x + 24 + 40 = 0 \)
Simplify to get:
\( x^2 + 16x + 64 = 0 \)

Step3: Solve the quadratic equation

Notice that \( x^2 + 16x + 64 \) is a perfect square trinomial, which factors as \( (x + 8)^2 = 0 \)
Taking the square root of both sides:
\( x + 8 = 0 \)
So, \( x = -8 \)

Step4: Find the corresponding \( y \)-value

Substitute \( x = -8 \) into the equation \( y = 18x - 40 \):
\( y = 18(-8) - 40 \)
\( y = -144 - 40 \)
\( y = -184 \)

Answer:

\( (-8, -184) \)