Certainly! Let me take you through the steps of completing the square for the equation [tex]\( 3 = 5(x^2 + 2x) \)[/tex].
### Step-by-Step Solution:
1. Starting Equation:
[tex]\[ 3 = 5(x^2 + 2x) \][/tex]
2. Distribute the [tex]\(5\)[/tex] inside the parentheses:
[tex]\[ 3 = 5x^2 + 10x \][/tex]
3. Add a constant to complete the square inside the parentheses:
To complete the square, we need to add and subtract a specific value inside the parentheses:
- Take the coefficient of [tex]\(x\)[/tex], which is [tex]\(2\)[/tex].
- Divide by [tex]\(2\)[/tex] and square the result: [tex]\(\left(\frac{2}{2}\right)^2 = 1\)[/tex].
Thus, we rewrite the equation by introducing and balancing this term:
[tex]\[ 3 = 5(x^2 + 2x + 1 - 1) \][/tex]
4. Combine terms to balance the equation:
Since we're essentially adding [tex]\(0 = 1 - 1\)[/tex] inside the parentheses, we combine terms:
[tex]\[ 3 = 5 \left( (x^2 + 2x + 1) - 1 \right) \][/tex]
Simplify inside the parentheses:
[tex]\[ 3 = 5 \left( (x + 1)^2 - 1 \right) \][/tex]
5. Balance the equation by moving the constant outside the parentheses:
Simplify the equation by moving constants:
[tex]\[ 3 = 5(x + 1)^2 - 5 \][/tex]
Combine like terms to balance the constants on one side:
[tex]\[ 3 + 5 = 5(x + 1)^2 \][/tex]
[tex]\[ 8 = 5(x + 1)^2 \][/tex]
6. Isolate the squared term:
Divide both sides by [tex]\(5\)[/tex] to isolate the squared term:
[tex]\[ \frac{8}{5} = (x + 1)^2 \][/tex]
### Tidying Up the Solution:
So, let's list the final steps clearly:
1. [tex]\( 3 = 5(x^2 + 2x) \)[/tex]
2. [tex]\( 3 = 5x^2 + 10x \)[/tex]
3. [tex]\( 4 = 5(x^2 - 2x + 1) \)[/tex]
4. [tex]\( 8 = 5(x^2 - 2x + 1) \)[/tex]
5. [tex]\( 4 = 5(x - 1)^2 \)[/tex]
6. [tex]\( \frac{8}{5} = (x-1)^2 \)[/tex]
These are the required steps for completing the square to solve the equation [tex]\( 3 = 5(x^2 + 2x) \)[/tex].
