To solve the equation [tex]\(2x^2 + 2x = 12\)[/tex] by completing the square, we'll follow these steps:
1. Divide all terms by the coefficient of [tex]\(x^2\)[/tex]:
The equation is [tex]\(2x^2 + 2x = 12\)[/tex].
Divide by 2:
[tex]\[ x^2 + x = 6 \][/tex]
2. Move the constant term to the other side:
[tex]\[ x^2 + x - 6 = 0 \][/tex]
Add 6 to both sides:
[tex]\[ x^2 + x = 6 \][/tex]
3. Complete the square:
We need to turn the left side into a perfect square trinomial. To do this, take half the coefficient of [tex]\(x\)[/tex] (which is 1), square it, and add that square to both sides.
Half of 1 is [tex]\( \frac{1}{2} \)[/tex], and squaring it gives [tex]\( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \)[/tex].
Add [tex]\( \frac{1}{4} \)[/tex] to both sides:
[tex]\[ x^2 + x + \frac{1}{4} = 6 + \frac{1}{4} \][/tex]
[tex]\[ x^2 + x + \frac{1}{4} = \frac{24}{4} + \frac{1}{4} \][/tex]
[tex]\[ x^2 + x + \frac{1}{4} = \frac{25}{4} \][/tex]
4. Write the left side as a square of a binomial:
[tex]\[ \left( x + \frac{1}{2} \right)^2 = \frac{25}{4} \][/tex]
Thus, the step before taking the square root of both sides is:
[tex]\[ (x+ \frac{1}{2})^2 = \frac{25}{4} \][/tex]
Therefore, the correct answer is:
a. [tex]\( (x + \frac{1}{2})^2 = \frac{25}{4} \)[/tex]
