Sure, let's go through the problem step-by-step and solve the equation [tex]\( x(6 + x) = 1 \)[/tex] by completing the square.
### Step 1: Expand and Rearrange the Equation
We start with the given equation:
[tex]\[ x(6 + x) = 1 \][/tex]
Expand the left side:
[tex]\[ 6x + x^2 = 1 \][/tex]
To make it look like a standard quadratic equation, rewrite it as:
[tex]\[ x^2 + 6x - 1 = 0 \][/tex]
### Step 2: Complete the Square
To solve this quadratic equation by completing the square, follow these steps:
1. Isolate the constant term on one side:
[tex]\[ x^2 + 6x = 1 \][/tex]
2. Find the value to complete the square:
Take the coefficient of [tex]\(x\)[/tex], which is 6, halve it, and then square it:
[tex]\[ \left(\frac{6}{2}\right)^2 = 9 \][/tex]
3. Add and subtract this square inside the equation:
[tex]\[ x^2 + 6x + 9 - 9 = 1 \][/tex]
4. Rewrite the left side as a perfect square and simplify:
[tex]\[ (x + 3)^2 - 9 = 1 \][/tex]
5. Isolate the perfect square by moving the constant:
[tex]\[ (x + 3)^2 = 10 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Take the square root on both sides:
[tex]\[ x + 3 = \pm \sqrt{10} \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = -3 \pm \sqrt{10} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = -3 + \sqrt{10} \][/tex]
[tex]\[ x_2 = -3 - \sqrt{10} \][/tex]
### Final Result
The solutions for the equation [tex]\( x(6 + x) = 1 \)[/tex] are:
[tex]\[ x_1 = -3 + \sqrt{10} \approx 0.162 \][/tex]
[tex]\[ x_2 = -3 - \sqrt{10} \approx -6.162 \][/tex]
Therefore, the solutions are [tex]\( x_1 \approx 0.162 \)[/tex] and [tex]\( x_2 \approx -6.162 \)[/tex].
