To solve the quadratic equation [tex]\(0 = x^2 - 6x + 4\)[/tex] by completing the square, let's go through the solution step-by-step:
1. Start with the original equation:
[tex]\[ 0 = x^2 - 6x + 4 \][/tex]
2. Rewrite the equation by isolating the quadratic and linear terms on one side:
[tex]\[ x^2 - 6x = -4 \][/tex]
3. To complete the square, we need to add and subtract the square of half the coefficient of [tex]\(x\)[/tex]. The coefficient of [tex]\(x\)[/tex] here is [tex]\(-6\)[/tex], so half of it is [tex]\(-3\)[/tex], and squaring it gives [tex]\(9\)[/tex]:
[tex]\[ x^2 - 6x + 9 = -4 + 9 \][/tex]
4. Now, our equation looks like this:
[tex]\[ (x - 3)^2 = 5 \][/tex]
5. Next, take the square root of both sides of the equation:
[tex]\[ \sqrt{(x - 3)^2} = \sqrt{5} \][/tex]
6. This gives us two possible equations since [tex]\((x - 3)\)[/tex] can be positive or negative:
[tex]\[ x - 3 = \sqrt{5} \][/tex]
[tex]\[ x - 3 = -\sqrt{5} \][/tex]
7. Solving these equations for [tex]\(x\)[/tex] gives:
[tex]\[ x = 3 + \sqrt{5} \][/tex]
[tex]\[ x = 3 - \sqrt{5} \][/tex]
Thus, the two solutions of the quadratic equation [tex]\(0 = x^2 - 6x + 4\)[/tex] are:
[tex]\[ x = 3 + \sqrt{5} \][/tex]
[tex]\[ x = 3 - \sqrt{5} \][/tex]
So the correct answer is:
[tex]\[ x = 3 + \sqrt{5} \text{ and } x = 3 - \sqrt{5} \][/tex]
