Final answer:
To solve the equation 2x^2 + 2x = 24, we complete the square to get (x + 1/2)^2 = 49/4, and find that x equals 3 or x equals -4 after simplifying and taking square roots.
Step-by-step explanation:
The value of x in the equation 2x^2 + 2x = 24 can be found by completing the square:
- First, move the constant term to the right side of the equation to get 2x^2 + 2x - 24 = 0.
- Divide all terms by 2 to simplify the equation: x^2 + x - 12 = 0.
- To complete the square, we need to add a number to both sides that makes the left side a perfect square trinomial. We calculate this number by taking half of the coefficient of x (1/2), squaring it, and adding it to both sides of the equation.
- The number to add is (1/2)^2 = 1/4. So, our equation becomes x^2 + x + 1/4 = 12 + 1/4 = 49/4.
- Now, the left side is a perfect square: (x + 1/2)^2 = 49/4. Taking the square root of both sides gives us x + 1/2 = ±√(49/4) which simplifies to x + 1/2 = ±7/2.
- Finally, subtracting 1/2 from both sides provides the solutions for x: x = 7/2 - 1/2 or x = -7/2 - 1/2. Therefore, x = 3 or x = -4.