How many real number solutions does 2x^2-5x+9 have?

 a
0

 b
1

 c
no real solutions

 d
3

To determine the number of real number solutions for the quadratic equation \(2x^2 - 5x + 9 = 0\), we can use the discriminant method. The discriminant \(D\) is given by:

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

For the quadratic equation \(ax^2 + bx + c = 0\), where:

- \(a = 2\)
- \(b = -5\)
- \(c = 9\)

Substitute the values into the discriminant formula:

\[
D = (-5)^2 - 4(2)(9)
\]

\[
D = 25 - 72
\]

\[
D = -47
\]

### Analyzing the Discriminant:
- If \(D > 0\), the equation has two distinct real solutions.
- If \(D = 0\), the equation has exactly one real solution (a repeated root).
- If \(D < 0\), the equation has no real solutions (the solutions are complex).

Since \(D = -47\) is less than 0, the quadratic equation \(2x^2 - 5x + 9 = 0\) has **no real solutions**.

### Final Answer:
- The correct answer is **(c) no real solutions**.

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Would you like more details or have any questions?

Here are some related questions:

1. How do you solve quadratic equations using the quadratic formula?
2. What does the discriminant tell us about the roots of a quadratic equation?
3. How do you find complex solutions for a quadratic equation?
4. Can a quadratic equation have more than two real solutions?
5. What is the geometric interpretation of the discriminant?
6. How can you determine the vertex of a quadratic equation?
7. What role does the leading coefficient play in a quadratic equation?
8. How do quadratic equations relate to parabolas?

**Tip:** The discriminant is a quick way to determine the nature of the roots without solving the entire equation.

Learn how to determine the number of real number solutions for the quadratic equation 2x^2 - 5x + 9 with a detailed explanation using the discriminant method. Discover when a quadratic equation has no real solutions and how to apply the discriminant formula effectively. Suitable for high school students.

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