Question

Solve the quadratic equation x^2 + 5x - 6 = 0 and provide step-by-step explanations for the solution

Answer 1

To solve the quadratic equation \( x^2 + 5x - 6 = 0 \), we can use the quadratic formula, which is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a \), \( b \), and \( c \) are coefficients from the equation \( ax^2 + bx + c = 0 \).

**Step 1: Identify coefficients.**

From the equation \( x^2 + 5x - 6 = 0 \):
- \( a = 1 \) (the coefficient of \( x^2 \))
- \( b = 5 \) (the coefficient of \( x \))
- \( c = -6 \) (the constant term)

**Step 2: Calculate the discriminant.**

The discriminant \( D \) is calculated as:

\[
D = b^2 - 4ac
\]

Plugging in the values of \( a \), \( b \), and \( c \):

\[
D = 5^2 - 4 \cdot 1 \cdot (-6)
\]
\[
D = 25 + 24
\]
\[
D = 49
\]

**Step 3: Calculate the roots using the quadratic formula.**

Since the discriminant is positive (\( D = 49 \)), we will have two distinct real roots. We use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]

Substituting the values of \( b \), \( D \), and \( a \):

\[
x = \frac{-5 \pm \sqrt{49}}{2 \cdot 1}
\]
\[
x = \frac{-5 \pm 7}{2}
\]

**Step 4: Solve for each root.**

Now we will compute the two possible values for \( x \):

1. **First root:**

\[
x_1 = \frac{-5 + 7}{2} = \frac{2}{2} = 1
\]

2. **Second root:**

\[
x_2 = \frac{-5 - 7}{2} = \frac{-12}{2} = -6
\]

**Step 5: State the solutions.**

The solutions to the equation \( x^2 + 5x - 6 = 0 \) are:

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

These provide us with the two values of \( x \) where the quadratic equation equals zero.