The given question asks us to solve the equation:
log³(x²−x−4)=2
First, rewrite the equation in exponential form:
3^2 = x² - x - 4
Simplify the equation:
x² - x - 4 - 9 = 0
x² - x - 13 = 0
This is a quadratic equation in standard form (ax² + bx + c = 0). To solve for x, we will need to use the quadratic formula: x = [-b ± sqrt(b² - 4ac)] / 2a.
For our equation, a = 1, b = -1, and c = -13.
Substitute these values into the quadratic formula:
x = [1 ± sqrt((-1)² - 4*1*(-13))] / 2*1
x = [1 ± sqrt(1 + 52)] / 2
x = [1 ± sqrt(53)] / 2
This gives us two possible exact solutions for x:
x =1/2 + sqrt(53)/2
x = 1/2 - sqrt(53)/2
These values are irrational, meaning they cannot be exactly represented as a finite decimal or a fraction. We can, however, approximate these solutions to three decimal places:
x ≈ 4.140
x ≈ -3.140
Therefore, the solutions to the equation log³(x²−x−4)=2 are approximately x = 4.140 and x = -3.140. Exact values will be x =1/2 + sqrt(53)/2 and x = 1/2 - sqrt(53)/2.