Question

solve the system of equations.
$y = x^2 - 15x + 50$
$y = -30x + 6$
write the coordinates in exact form. simplify all fractions and radicals.

Explanation:

Step1: Set the two equations equal

Since both equations equal \( y \), we set them equal to each other: \( x^2 - 15x + 50 = -30x + 6 \)

Step2: Rearrange into standard quadratic form

Move all terms to the left side: \( x^2 - 15x + 30x + 50 - 6 = 0 \) simplifies to \( x^2 + 15x + 44 = 0 \)

Step3: Factor the quadratic (or use quadratic formula)

Factor: \( (x + 4)(x + 11) = 0 \)

Step4: Solve for \( x \)

Set each factor to zero: \( x + 4 = 0 \) gives \( x = -4 \); \( x + 11 = 0 \) gives \( x = -11 \)

Step5: Find corresponding \( y \) values

For \( x = -4 \): \( y = -30(-4) + 6 = 120 + 6 = 126 \)
For \( x = -11 \): \( y = -30(-11) + 6 = 330 + 6 = 336 \)

Answer:

\((-4, 126)\), \((-11, 336)\)