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Answer :
To solve the quadratic equation [tex]\((12x - 27)^2 = 256\)[/tex], follow these steps:
1. Take the square root of both sides: The equation is [tex]\((12x - 27)^2 = 256\)[/tex]. To eliminate the square, take the square root on both sides:
[tex]\[
12x - 27 = \pm \sqrt{256}
\][/tex]
2. Calculate the square root: [tex]\(\sqrt{256} = 16\)[/tex]. So, the equation becomes:
[tex]\[
12x - 27 = 16 \quad \text{or} \quad 12x - 27 = -16
\][/tex]
3. Solve each equation separately:
For [tex]\(12x - 27 = 16\)[/tex]:
- Add 27 to both sides: [tex]\(12x = 16 + 27\)[/tex]
- Simplify: [tex]\(12x = 43\)[/tex]
- Divide by 12: [tex]\(x = \frac{43}{12}\)[/tex]
For [tex]\(12x - 27 = -16\)[/tex]:
- Add 27 to both sides: [tex]\(12x = -16 + 27\)[/tex]
- Simplify: [tex]\(12x = 11\)[/tex]
- Divide by 12: [tex]\(x = \frac{11}{12}\)[/tex]
4. Write the solutions: The solutions to the quadratic equation are:
[tex]\[
x = \frac{43}{12} \quad \text{and} \quad x = \frac{11}{12}
\][/tex]
Thus, the correct answer is option C: [tex]\(x = \frac{43}{12} ; -\frac{11}{12}\)[/tex].
1. Take the square root of both sides: The equation is [tex]\((12x - 27)^2 = 256\)[/tex]. To eliminate the square, take the square root on both sides:
[tex]\[
12x - 27 = \pm \sqrt{256}
\][/tex]
2. Calculate the square root: [tex]\(\sqrt{256} = 16\)[/tex]. So, the equation becomes:
[tex]\[
12x - 27 = 16 \quad \text{or} \quad 12x - 27 = -16
\][/tex]
3. Solve each equation separately:
For [tex]\(12x - 27 = 16\)[/tex]:
- Add 27 to both sides: [tex]\(12x = 16 + 27\)[/tex]
- Simplify: [tex]\(12x = 43\)[/tex]
- Divide by 12: [tex]\(x = \frac{43}{12}\)[/tex]
For [tex]\(12x - 27 = -16\)[/tex]:
- Add 27 to both sides: [tex]\(12x = -16 + 27\)[/tex]
- Simplify: [tex]\(12x = 11\)[/tex]
- Divide by 12: [tex]\(x = \frac{11}{12}\)[/tex]
4. Write the solutions: The solutions to the quadratic equation are:
[tex]\[
x = \frac{43}{12} \quad \text{and} \quad x = \frac{11}{12}
\][/tex]
Thus, the correct answer is option C: [tex]\(x = \frac{43}{12} ; -\frac{11}{12}\)[/tex].
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