9. Find the solutions of the system algebraically.

Question

y = x*2 − 2x − 6
y= 4x + 10

A. (2, -2) and (8, 42)
B. (-2, 2) and (-8, -42)
C. (-2, 2) and (8, 42)
D. no solution

Answer 1

To find the solution of the system algebraically, we can set the two equations equal to each other:

x^2 − 2x − 6 = 4x + 10

Rearranging terms, we get:

x^2 - 6x - 16 = 0

Factoring this quadratic equation, we get:

(x - 8)(x + 2) = 0

Setting each factor to zero:

x - 8 = 0
x = 8

x + 2 = 0
x = -2

Now, substitute these x-values back into one of the original equations to find the corresponding y-values:

For x = 8:
y = 8^2 - 2(8) - 6
y = 42

For x = -2:
y = (-2)^2 - 2(-2) - 6
y = 2

Therefore, the solutions to the system are (8, 42) and (-2, 2), which matches option C.