Solve the quadratic equation by using the quadratic formula. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.)
2x2 − 5x − 1 = 0


The given quadratic equation is:

\[
2x^2 - 5x - 1 = 0
\]

The quadratic formula is:

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

For the equation \(2x^2 - 5x - 1 = 0\), the coefficients are:
- \(a = 2\)
- \(b = -5\)
- \(c = -1\)

Now, we can substitute these values into the quadratic formula:

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

Simplifying step by step:

1. Calculate the discriminant:
   \[
   (-5)^2 - 4(2)(-1) = 25 + 8 = 33
   \]

2. Substitute the discriminant and simplify:
   \[
   x = \frac{5 \pm \sqrt{33}}{4}
   \]

Therefore, the solutions are:

\[
x = \frac{5 + \sqrt{33}}{4}, \quad x = \frac{5 - \sqrt{33}}{4}
\]

Since these are the exact solutions, you can also approximate them:

\[
x \approx \frac{5 + 5.744}{4} \approx \frac{10.744}{4} \approx 2.686
\]
\[
x \approx \frac{5 - 5.744}{4} \approx \frac{-0.744}{4} \approx -0.186
\]

Thus, the solutions are approximately:

\[
x \approx 2.686, -0.186
\]

Would you like further details or clarifications?

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Here are 5 related questions for further exploration:

1. What is the significance of the discriminant in determining the nature of the roots?
2. How can you complete the square to solve a quadratic equation?
3. What is the relationship between the quadratic formula and the axis of symmetry of a parabola?
4. How can you graphically interpret the roots of a quadratic equation?
5. What happens if the discriminant is negative in the quadratic formula?

**Tip:** Always check the discriminant first to know if real or complex solutions exist before solving.

Solve the quadratic equation by using the quadratic formula. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) 2x^2 − 5x − 1 = 0

Learn how to solve the quadratic equation 2x^2 - 5x - 1 using the quadratic formula with step-by-step guidance. This tutorial covers key algebraic concepts, including how to compute the discriminant and approximate the roots. Suitable for students in Grades 9-11.

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