Let's solve the equation [tex]\(\log_2(x^2 - x - 2) = 2\)[/tex] step-by-step.
### Step 1: Convert Logarithmic Equation to Exponential Form
The given equation is [tex]\(\log_2(x^2 - x - 2) = 2\)[/tex]. We know that if [tex]\(\log_b(a) = c\)[/tex], then [tex]\(b^c = a\)[/tex].
So, converting [tex]\(\log_2(x^2 - x - 2) = 2\)[/tex] to its exponential form, we get:
[tex]\[
x^2 - x - 2 = 2^2
\][/tex]
### Step 2: Simplify the Equation
We know that [tex]\(2^2 = 4\)[/tex]. Substituting this back into the equation:
[tex]\[
x^2 - x - 2 = 4
\][/tex]
Next, we move all terms to one side of the equation to set it to zero:
[tex]\[
x^2 - x - 2 - 4 = 0
\][/tex]
[tex]\[
x^2 - x - 6 = 0
\][/tex]
### Step 3: Solve the Quadratic Equation
We have the quadratic equation [tex]\(x^2 - x - 6 = 0\)[/tex]. To solve this, we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation [tex]\(x^2 - x - 6 = 0\)[/tex], the coefficients are [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -6\)[/tex].
First, calculate the discriminant:
[tex]\[
\text{discriminant} = b^2 - 4ac
\][/tex]
Substitute the values:
[tex]\[
\text{discriminant} = (-1)^2 - 4(1)(-6) = 1 + 24 = 25
\][/tex]
Next, compute the square root of the discriminant:
[tex]\[
\sqrt{\text{discriminant}} = \sqrt{25} = 5
\][/tex]
Now, apply the quadratic formula:
[tex]\[
x = \frac{-(-1) \pm 5}{2(1)}
\][/tex]
[tex]\[
x = \frac{1 \pm 5}{2}
\][/tex]
This yields two solutions:
[tex]\[
x_1 = \frac{1 + 5}{2} = \frac{6}{2} = 3
\][/tex]
[tex]\[
x_2 = \frac{1 - 5}{2} = \frac{-4}{2} = -2
\][/tex]
### Step 4: Verify the Solutions
We should quickly verify that both solutions satisfy the original equation. Substitute [tex]\(x = 3\)[/tex] and [tex]\(x = -2\)[/tex] back into the original logarithmic equation [tex]\(\log_2(x^2 - x - 2)\)[/tex]:
For [tex]\(x = 3\)[/tex]:
[tex]\[
3^2 - 3 - 2 = 9 - 3 - 2 = 4 \quad \Rightarrow \quad \log_2(4) = 2 \quad \text{(true)}
\][/tex]
For [tex]\(x = -2\)[/tex]:
[tex]\[
(-2)^2 - (-2) - 2 = 4 + 2 - 2 = 4 \quad \Rightarrow \quad \log_2(4) = 2 \quad \text{(true)}
\][/tex]
Thus, both [tex]\(x = 3\)[/tex] and [tex]\(x = -2\)[/tex] are valid solutions.
### Summary
The solutions to the given equation [tex]\(\log_2(x^2 - x - 2) = 2\)[/tex] are:
[tex]\[
x = 3 \quad \text{and} \quad x = -2
\][/tex]
