Use the discriminant to determine the number and nature of the solutions to the quadratic equation. Do not solve the equation.

9 x squared plus 2 x minus 10 equals 09x2+2x−10=0

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Part 1

The discriminant is equal to 

enter your response here.
 
​(Type an integer or a simplified​ fraction.)

To determine the number and nature of the solutions of a quadratic equation using the discriminant, we start with the standard form of a quadratic equation:

\[
ax^2 + bx + c = 0
\]

The discriminant (\(D\)) of a quadratic equation is given by the formula:

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

For the given quadratic equation:

\[
9x^2 + 2x - 10 = 0
\]

we can identify the coefficients as:

\[
a = 9, \quad b = 2, \quad c = -10
\]

Now, plug these values into the formula for the discriminant:

\[
D = (2)^2 - 4(9)(-10)
\]

Simplify the expression:

\[
D = 4 + 360 = 364
\]

### Conclusion:
The discriminant is equal to **364**.

Since the discriminant is positive and not a perfect square, the quadratic equation has **two distinct real and irrational solutions**.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the geometric interpretation of the discriminant?
2. How does the discriminant change if the coefficient of \(x^2\) is modified?
3. What happens if the discriminant is zero?
4. Can the discriminant tell us if the solutions are rational or irrational?
5. How does the value of the discriminant affect the graph of the quadratic function?

**Tip:** Remember, the discriminant gives insight into the nature of the roots without actually solving the equation!

Use the discriminant to determine the number and nature of the solutions to the quadratic equation. Do not solve the equation. 9x^2 + 2x - 10 = 0.

Explore how to use the discriminant to determine the number and nature of the solutions for the quadratic equation 9x^2 + 2x - 10. This tutorial covers key algebraic concepts such as the discriminant formula and its application. Ideal for students in Grades 8-10.

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