Certainly! Let's solve the equation step-by-step.
We start with the given equation:
[tex]\[
\frac{x^2 - (x + 1)(x - 2)}{5} - 6 = 0
\][/tex]
Step 1: Simplify the expression inside the numerator.
[tex]\[
x^2 - (x + 1)(x - 2)
\][/tex]
First, expand the term [tex]\((x + 1)(x - 2)\)[/tex]:
[tex]\[
(x + 1)(x - 2) = x^2 - 2x + x - 2 = x^2 - x - 2
\][/tex]
Now, subtract this from [tex]\(x^2\)[/tex]:
[tex]\[
x^2 - (x^2 - x - 2) = x^2 - x^2 + x + 2 = x + 2
\][/tex]
So the expression becomes:
[tex]\[
\frac{x + 2}{5} - 6 = 0
\][/tex]
Step 2: Move 6 to the right side of the equation:
[tex]\[
\frac{x + 2}{5} = 6
\][/tex]
Step 3: Eliminate the fraction by multiplying both sides by 5:
[tex]\[
x + 2 = 30
\][/tex]
Step 4: Isolate x by subtracting 2 from both sides:
[tex]\[
x = 28
\][/tex]
Our solution is:
[tex]\[
x = 28
\][/tex]
Step 5: Ensure that our solution does not violate the given condition [tex]\(x \neq 1\)[/tex]. Since [tex]\(x = 28\)[/tex] does not violate this condition, it is a valid solution.
Hence, the solution to the equation is:
[tex]\[
x = 28
\][/tex]
