IDNLearn.com: Your go-to resource for finding expert answers. Find the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
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]
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]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.