Solve the quadratic equation.

5x2 - 2x -1 = 0

To solve the quadratic equation \(5x^2 - 2x - 1 = 0\), we can apply the **quadratic formula**:

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

where \(a = 5\), \(b = -2\), and \(c = -1\).

### Step 1: Plug the coefficients into the formula

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(5)(-1)}}{2(5)}
\]

Simplify:

\[
x = \frac{2 \pm \sqrt{4 + 20}}{10}
\]

\[
x = \frac{2 \pm \sqrt{24}}{10}
\]

### Step 2: Simplify the square root

\[
\sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6}
\]

So:

\[
x = \frac{2 \pm 2\sqrt{6}}{10}
\]

### Step 3: Simplify the fraction

Factor out 2 from the numerator:

\[
x = \frac{2(1 \pm \sqrt{6})}{10}
\]

\[
x = \frac{1 \pm \sqrt{6}}{5}
\]

### Final Answer

The two solutions are:

\[
x_1 = \frac{1 + \sqrt{6}}{5}, \quad x_2 = \frac{1 - \sqrt{6}}{5}
\]

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Would you like more details, or do you have any questions?

Here are 5 related questions you might find helpful:
1. How do you determine the discriminant of a quadratic equation?
2. What do real and imaginary solutions mean for quadratic equations?
3. Can you solve the same equation by completing the square?
4. How does the quadratic formula handle repeated roots?
5. What are some applications of quadratic equations in real-world problems?

**Tip:** Always check the discriminant \(b^2 - 4ac\) to determine the nature of the roots (real or complex).

Solve the quadratic equation: 5x^2 - 2x - 1 = 0

Learn how to solve the quadratic equation 5x^2 - 2x - 1 = 0 step by step using the quadratic formula. This solution covers core algebraic concepts, such as the discriminant and square root simplification, suitable for Grades 8-10 students.

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