To solve the equation \( x^4 - 5x^2 - 36 = 0 \) using factoring, we can proceed as follows:
1. Rewrite the equation: Start with the original equation:
[tex]\[
x^4 - 5x^2 - 36 = 0
\][/tex]
2. Substitute \( y = x^2 \): This substitution simplifies the polynomial:
[tex]\[
y^2 - 5y - 36 = 0
\][/tex]
Here, \( y = x^2 \).
3. Factor the quadratic equation: We need to factor \( y^2 - 5y - 36 = 0 \). We look for two numbers that multiply to \(-36\) and add to \(-5\). These numbers are \(-9\) and \(4\):
[tex]\[
y^2 - 5y - 36 = (y - 9)(y + 4) = 0
\][/tex]
4. Solve for \( y \): Set each factor equal to zero to solve for \( y \):
[tex]\[
y - 9 = 0 \quad \Rightarrow \quad y = 9
\][/tex]
[tex]\[
y + 4 = 0 \quad \Rightarrow \quad y = -4
\][/tex]
5. Back-substitute \( y = x^2 \): Replace \( y \) with \( x^2 \):
- For \( y = 9 \):
[tex]\[
x^2 = 9 \quad \Rightarrow \quad x = \pm 3
\][/tex]
- For \( y = -4 \):
[tex]\[
x^2 = -4 \quad \Rightarrow \quad x = \pm 2i
\][/tex]
6. List the solutions: The solutions to the equation \( x^4 - 5x^2 - 36 = 0 \) are:
[tex]\[
x = \pm 3, \pm 2i
\][/tex]
Thus, the correct answer is:
[tex]\[ x = \pm 2i \text{ and } x = \pm 3 \][/tex]
