use the quadratic formula to find x: x^2+2x-5=0

To solve the quadratic equation \(x^2 + 2x - 5 = 0\) using the quadratic formula, follow these steps:

The quadratic formula is given by:

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

For the equation \(x^2 + 2x - 5 = 0\), the coefficients are:
- \(a = 1\) (coefficient of \(x^2\)),
- \(b = 2\) (coefficient of \(x\)),
- \(c = -5\) (constant term).

### Step 1: Calculate the discriminant.
The discriminant is:

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

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

\[
\Delta = 2^2 - 4(1)(-5) = 4 + 20 = 24
\]

### Step 2: Apply the quadratic formula.
Now substitute the values of \(a\), \(b\), and \(\Delta\) into the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{24}}{2(1)} = \frac{-2 \pm \sqrt{24}}{2}
\]

Since \(\sqrt{24} = 2\sqrt{6}\), the equation becomes:

\[
x = \frac{-2 \pm 2\sqrt{6}}{2}
\]

### Step 3: Simplify the result.

\[
x = -1 \pm \sqrt{6}
\]

Thus, the two solutions are:

\[
x_1 = -1 + \sqrt{6}, \quad x_2 = -1 - \sqrt{6}
\]

### Final answer:

\[
x_1 = -1 + \sqrt{6}, \quad x_2 = -1 - \sqrt{6}
\]

Would you like more details on any step or further explanation?

Here are some related questions:
1. How do you find the discriminant of a quadratic equation?
2. What does the discriminant tell you about the roots of a quadratic equation?
3. Can all quadratic equations be solved using the quadratic formula?
4. How would you simplify radical expressions like \(\sqrt{24}\)?
5. What are some alternative methods for solving quadratic equations?

**Tip**: Always check the discriminant first to understand the nature of the roots (real vs. complex).

Learn how to solve the quadratic equation x^2 + 2x - 5 = 0 using the quadratic formula. This step-by-step solution explains the process of calculating the discriminant and simplifying the equation. Perfect for students in Grades 8-10.

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