To solve the quadratic equation [tex]\(x^2 - 6x + 4 = 0\)[/tex] by completing the square, follow these steps:
1. Start with the given equation:
[tex]\[
x^2 - 6x + 4 = 0
\][/tex]
2. Move the constant term to the right side:
[tex]\[
x^2 - 6x = -4
\][/tex]
3. Complete the square on the left side:
- Take half of the coefficient of [tex]\(x\)[/tex] (which is [tex]\(-6\)[/tex]), divide by 2, and square it:
[tex]\[
\left(\frac{-6}{2}\right)^2 = 9
\][/tex]
- Add and subtract this value to/from the left side:
[tex]\[
x^2 - 6x + 9 - 9 = -4
\][/tex]
- Rearrange the equation:
[tex]\[
(x - 3)^2 - 9 = -4
\][/tex]
4. Move the constant term [tex]\(-9\)[/tex] to the right side to isolate the square term:
[tex]\[
(x - 3)^2 = 5
\][/tex]
5. Take the square root of both sides:
[tex]\[
x - 3 = \pm \sqrt{5}
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = 3 + \sqrt{5} \quad \text{or} \quad x = 3 - \sqrt{5}
\][/tex]
So, the solutions for the quadratic equation [tex]\(x^2 - 6x + 4 = 0\)[/tex] are:
[tex]\[
x = 3 + \sqrt{5} \quad \text{and} \quad x = 3 - \sqrt{5}
\][/tex]
