To solve the given equation [tex]\(x^2 - 4x + 4 = 2x\)[/tex], we need to go through a few steps. Here’s a detailed, step-by-step solution:
### Step 1: Rewrite the equation
First, move all terms to one side of the equation to set it equal to zero:
[tex]\[
x^2 - 4x + 4 - 2x = 0
\][/tex]
Combine like terms:
[tex]\[
x^2 - 6x + 4 = 0
\][/tex]
Now, we'll rewrite the quadratic equation by completing the square.
### Completing the Square
To complete the square, we follow these steps:
1. Take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-6\)[/tex], divide it by 2, and square it:
[tex]\[
\left(\frac{-6}{2}\right)^2 = 9
\][/tex]
2. Add and subtract this square term inside the equation:
[tex]\[
x^2 - 6x + 9 - 9 + 4 = 0
\][/tex]
This can be rewritten as:
[tex]\[
(x - 3)^2 - 5 = 0
\][/tex]
So the equation in completed square form is:
[tex]\[
(x - 3)^2 = 5
\][/tex]
### Step 2: Find the solutions to the equation
The equation [tex]\((x - 3)^2 = 5\)[/tex] indicates that the solutions are found by taking the square root of both sides. This gives us:
[tex]\[
x - 3 = \pm\sqrt{5}
\][/tex]
Thus, we have two potential solutions:
[tex]\[
x - 3 = \sqrt{5}
\][/tex]
or
[tex]\[
x - 3 = -\sqrt{5}
\][/tex]
Solving for [tex]\(x\)[/tex] in both cases:
1. For [tex]\(x - 3 = \sqrt{5}\)[/tex]:
[tex]\[
x = 3 + \sqrt{5}
\][/tex]
2. For [tex]\(x - 3 = -\sqrt{5}\)[/tex]:
[tex]\[
x = 3 - \sqrt{5}
\][/tex]
### Numerical Solutions
Providing the numerical solutions for clarity:
[tex]\[
x \approx 3 + 2.236 = 5.236
\][/tex]
and
[tex]\[
x \approx 3 - 2.236 = 0.764
\][/tex]
So, the solutions to the equation [tex]\(x^2 - 4x + 4 = 2x\)[/tex] are approximately [tex]\(x \approx 5.236\)[/tex] and [tex]\(x \approx 0.764\)[/tex].
