To find all solutions, real or imaginary, to the equation [tex]\(4(x - 1)^2 + 8 = 0\)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[
4(x - 1)^2 + 8 = 0
\][/tex]
2. Isolate the quadratic term:
Subtract 8 from both sides of the equation:
[tex]\[
4(x - 1)^2 = -8
\][/tex]
3. Divide both sides by 4:
[tex]\[
(x - 1)^2 = -2
\][/tex]
4. Take the square root of both sides:
To solve for [tex]\(x\)[/tex], take the square root of both sides:
[tex]\[
x - 1 = \pm \sqrt{-2}
\][/tex]
Recall that the square root of a negative number introduces the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
x - 1 = \pm \sqrt{2}i
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Add 1 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x = 1 \pm \sqrt{2}i
\][/tex]
6. Write the solutions:
The solutions can be expressed as:
[tex]\[
x = 1 - \sqrt{2}i \quad \text{and} \quad x = 1 + \sqrt{2}i
\][/tex]
Therefore, the solutions to the equation [tex]\(4(x - 1)^2 + 8 = 0\)[/tex] are:
[tex]\[
x = 1 - \sqrt{2}i \quad \text{and} \quad x = 1 + \sqrt{2}i
\][/tex]
