Final answer:
The solutions to the equation 2(x + 6)(x−1) =12 are x = (-10 + 2sqrt(73))/4 and x = (-10 - 2sqrt(73))/4.
Step-by-step explanation:
To solve the equation 2(x + 6) (x−1) =12, we need to expand and simplify the equation first:
2(x + 6)(x−1) = 12
2(x^2 + 5x - 6) = 12
2x^2 + 10x - 12 = 12
2x^2 + 10x - 24 = 0
Next, we can use the quadratic formula to find the solutions for x. The quadratic formula states: x = (-b ± sqrt(b^2 - 4ac))/(2a)
For our equation, a = 2, b = 10, and c = -24. Substituting these values into the quadratic formula, we get:
x = (-10 ± sqrt(10^2 - 4(2)(-24)))/(2(2))
Simplifying further, we have:
x = (-10 ± sqrt(100 + 192))/4
x = (-10 ± sqrt(292))/4
Finally, we can simplify under the square root:
x = (-10 ± sqrt(4*73))/4
Now we can split the calculation into two parts:
x = (-10 ± 2sqrt(73))/4
Therefore, the solutions to the equation 2(x + 6)(x−1) =12 are x = (-10 + 2sqrt(73))/4 and x = (-10 - 2sqrt(73))/4.