Final answer:
To solve for x, set r(x) = k(x) and use the quadratic formula to find the solutions.
Step-by-step explanation:
We can set r(x) = k(x) and solve for x.
Given:
r(x) = 4x^(2) + 4x - 2
k(x) = -2x^(2) - 3x + 8
Set r(x) = k(x):
4x^(2) + 4x - 2 = -2x^(2) - 3x + 8
Combine like terms:
6x^(2) + 7x - 10 = 0
Now we can use the quadratic formula to find the solutions for x.
Quadratic formula: x = (-b ± sqrt(b^(2) - 4ac))/(2a)
Substitute the values from our equation:
x = (-7 ± sqrt(7^(2) - 4 * 6 * (-10)))/(2 * 6)
Simplify the expression under the square root:
x = (-7 ± sqrt(49 + 240))/(12)
x = (-7 ± sqrt(289))/(12)
Take the square root:
x = (-7 ± 17)/(12)
So, the two values of x that make r(x) = k(x) are x = 10/6 and x = -24/6.