Sure! Let's solve the problem step by step.
First, we start with the given equation:
[tex]\[ x^2 - 6 = 2 - 18x \][/tex]
### Step 1: Rewrite the equation in standard form
To rewrite this equation in a standard form [tex]\(ax^2 + bx + c = 0\)[/tex], we need to move everything to one side of the equation:
[tex]\[ x^2 - 6 - 2 + 18x = 0 \][/tex]
which simplifies to:
[tex]\[ x^2 + 18x - 8 = 0 \][/tex]
### Step 2: Complete the square
To complete the square, we focus on the quadratic and linear terms [tex]\(x^2 + 18x\)[/tex]. We add and subtract a term to make it a perfect square trinomial. The process is as follows:
1. Take the coefficient of [tex]\(x\)[/tex], which is 18, divide it by 2, and square it: [tex]\((\frac{18}{2})^2 = 81\)[/tex].
2. Add and subtract this square inside the equation:
[tex]\[ x^2 + 18x - 8 = 0 \implies x^2 + 18x + 81 - 81 - 8 = 0 \][/tex]
[tex]\[ (x + 9)^2 - 81 - 8 = 0 \][/tex]
3. Simplify:
[tex]\[ (x + 9)^2 - 89 = 0 \][/tex]
Thus, the completed square form of the equation is:
[tex]\[ (x + 9)^2 = 89 \][/tex]
Here, [tex]\(c = 9\)[/tex] and [tex]\(d = 89\)[/tex].
### Step 3: Solve the equation for [tex]\(x\)[/tex]
To solve the equation [tex]\((x + 9)^2 = 89\)[/tex], we take the square root of both sides:
[tex]\[ x + 9 = \pm \sqrt{89} \][/tex]
This gives us two solutions:
[tex]\[ x = -9 + \sqrt{89} \][/tex]
[tex]\[ x = -9 - \sqrt{89} \][/tex]
### Conclusion
The solutions to the equation are:
[tex]\[ x = -9 + \sqrt{89} \][/tex]
[tex]\[ x = -9 - \sqrt{89} \][/tex]
Therefore, the correct answer is:
[tex]\[ B) x = -9 \pm \sqrt{89} \][/tex]
