Question

solve the system of equations.
y = x² - 18x - 45
y = -5x - 15
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: \( x^2 - 18x - 45 = -5x - 15 \)

Step2: Rearrange into standard quadratic form

Move all terms to the left side: \( x^2 - 18x + 5x - 45 + 15 = 0 \)
Simplify: \( x^2 - 13x - 30 = 0 \)

Step3: Solve the quadratic equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -13 \), \( c = -30 \)
Calculate the discriminant: \( \Delta = (-13)^2 - 4(1)(-30) = 169 + 120 = 289 \)
Then \( x = \frac{13 \pm \sqrt{289}}{2(1)} = \frac{13 \pm 17}{2} \)

Step4: Find the two x-values

First solution: \( x = \frac{13 + 17}{2} = \frac{30}{2} = 15 \)
Second solution: \( x = \frac{13 - 17}{2} = \frac{-4}{2} = -2 \)

Step5: Find the corresponding y-values

For \( x = 15 \): \( y = -5(15) - 15 = -75 - 15 = -90 \)
For \( x = -2 \): \( y = -5(-2) - 15 = 10 - 15 = -5 \)

Answer:

\( (15, -90) \)
\( (-2, -5) \)