Question

Solve for x in the quadratic equation x^2 + 5x - 6 = 0 and provide a step-by-step explanation.

Answer 1

To solve the quadratic equation \( x^2 + 5x - 6 = 0 \) step by step, we can use the factoring method.

### Step 1: Identify and Rewrite the Equation
We start with the equation:
\[
x^2 + 5x - 6 = 0
\]

### Step 2: Factor the Quadratic Expression
We need to factor the expression \( x^2 + 5x - 6 \). To do this, we look for two numbers that multiply to \( -6 \) (the constant term) and add to \( 5 \) (the coefficient of the \( x \) term).

The pairs of factors of \( -6 \) are:
- \( (-1, 6) \)
- \( (1, -6) \)
- \( (-2, 3) \)
- \( (2, -3) \)

Among these pairs, we notice that \( -1 \) and \( 6 \) multiply to \( -6 \) and add to \( 5 \):
\[
-1 + 6 = 5
\]

### Step 3: Write the Factors
Using these two numbers, we can factor the quadratic expression:
\[
x^2 + 5x - 6 = (x - 1)(x + 6)
\]

### Step 4: Set Each Factor to Zero
Next, we set each factor equal to zero to find the values of \( x \):
\[
x - 1 = 0 \quad \text{or} \quad x + 6 = 0
\]

### Step 5: Solve for \( x \)
Now, solve each equation:
1. For \( x - 1 = 0 \):
\[
x = 1
\]

2. For \( x + 6 = 0 \):
\[
x = -6
\]

### Step 6: Write the Solution
Thus, the solutions to the quadratic equation \( x^2 + 5x - 6 = 0 \) are:
\[
x = 1 \quad \text{and} \quad x = -6
\]

### Summary
The solutions are:
\[
\boxed{1} \quad \text{and} \quad \boxed{-6}
\]