Solve the Quadratic Inequality: x²-2x-8>0 Step by Step

To solve this quadratic inequality $x^2 - 2x - 8 > 0$, we will follow these steps:

• Step 1: Factor the quadratic expression.
• Step 2: Determine the roots of the equation.
• Step 3: Analyze the sign of the quadratic expression over the different intervals formed by these roots.

Step 1: We start by factoring $x^2 - 2x - 8$. The expression factors as follows:

$x^2 - 2x - 8 = (x - 4)(x + 2)$

Step 2: Set each factor to zero to find the roots:

• $x - 4 = 0$ which gives $x = 4$.
• $x + 2 = 0$ which gives $x = -2$.

Step 3: These roots $x = -2$ and $x = 4$ divide the real number line into three intervals: $x < -2$, $-2 < x < 4$, and $x > 4$. We will test each interval to determine where the inequality holds true:

• For $x < -2$, choose a test point, say $x = -3$. Substitute into the factored expression: $(x - 4)(x + 2) = (-3 - 4)(-3 + 2) = (-7)(-1) = 7$. This is positive, so the inequality holds.
• For $-2 < x < 4$, choose a test point, say $x = 0$. Substitute into the factored expression: $(0 - 4)(0 + 2) = (-4)(2) = -8$. This is negative, so the inequality does not hold.
• For $x > 4$, choose a test point, say $x = 5$. Substitute into the factored expression: $(5 - 4)(5 + 2) = (1)(7) = 7$. This is positive, so the inequality holds.

Thus, the solution to the inequality $x^2 - 2x - 8 > 0$ is $x < -2$ or $x > 4$.

Comparing with the given choices, the correct answer is:

Answers (a) and (c)