Question

How to solve the quadratic equation x^2 + 5x - 6 = 0?

Answer 1

To solve the quadratic equation \( x^2 + 5x - 6 = 0 \), we can use the factoring method, the quadratic formula, or complete the square. Here, we will first try factoring.

### Step 1: Factor the quadratic
We need to express the quadratic in the form \( (x + a)(x + b) = 0 \), where \( a \) and \( b \) are two numbers that multiply to \(-6\) (the constant term) and add to \(5\) (the coefficient of \(x\)).

The numbers that work are \(6\) and \(-1\), since:

- \(6 + (-1) = 5\)
- \(6 \times (-1) = -6\)

Thus, we can factor the quadratic:

\[
(x + 6)(x - 1) = 0
\]

### Step 2: Set each factor equal to zero
Now we can set each factor equal to zero and solve for \(x\):

1. \(x + 6 = 0\)
\[
x = -6
\]

2. \(x - 1 = 0\)
\[
x = 1
\]

### Step 3: Solutions
The solutions to the quadratic equation \(x^2 + 5x - 6 = 0\) are:

\[
x = -6 \quad \text{and} \quad x = 1
\]

### Verification
To verify, we can substitute the solutions back into the original equation:

For \(x = -6\):
\[
(-6)^2 + 5(-6) - 6 = 36 - 30 - 6 = 0
\]

For \(x = 1\):
\[
(1)^2 + 5(1) - 6 = 1 + 5 - 6 = 0
\]

Both solutions satisfy the equation, confirming that they are correct.

### Final Answer
The solutions are:
\[
x = -6 \quad \text{and} \quad x = 1
\]