Question

Solve the quadratic by factoring.
x, squared, minus, 4, x, minus, 41, equals, 4
x
2
−4x−41=4

Answer 1

To solve the quadratic equation \( x^2 - 4x - 41 = 4 \) by factoring, we first need to set the equation to zero:

1. Subtract 4 from both sides:
\[
x^2 - 4x - 41 - 4 = 0
\]
This simplifies to:
\[
x^2 - 4x - 45 = 0
\]

2. Now, we factor the quadratic \( x^2 - 4x - 45 \). We need to find two numbers that multiply to -45 (the constant term) and add to -4 (the coefficient of \( x \)).

The two numbers that satisfy these conditions are \( -9 \) and \( 5 \) because:
- \( -9 \times 5 = -45 \)
- \( -9 + 5 = -4 \)

3. We can write the factored form as:
\[
(x - 9)(x + 5) = 0
\]

4. Now, we set each factor to zero and solve for \( x \):
\[
x - 9 = 0 \quad \Rightarrow \quad x = 9
\]
\[
x + 5 = 0 \quad \Rightarrow \quad x = -5
\]

5. Therefore, the solutions to the equation \( x^2 - 4x - 41 = 4 \) are:
\[
x = 9 \quad \text{and} \quad x = -5
\]

In conclusion, the solutions are:
\[
x = 9 \quad \text{and} \quad x = -5
\]