Determine the number and type of solutions for the equation 5x^2+5x-3+0

The equation given is \( 5x^2 + 5x - 3 = 0 \).

To determine the number and type of solutions for a quadratic equation of the form \( ax^2 + bx + c = 0 \), we use the discriminant (\(\Delta\)):

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

For the given equation \( 5x^2 + 5x - 3 = 0 \):

- \( a = 5 \)
- \( b = 5 \)
- \( c = -3 \)

Substituting these values into the discriminant formula:

\[
\Delta = 5^2 - 4(5)(-3) = 25 + 60 = 85
\]

Since the discriminant (\(\Delta\)) is positive and not a perfect square:

1. **Number of Solutions**: There are **two distinct real solutions**.
2. **Type of Solutions**: The solutions are **irrational** because the discriminant is not a perfect square.

Would you like more details on how to find the exact solutions? 

Here are 5 questions related to this topic:
1. How do you calculate the exact solutions using the quadratic formula?
2. What are the conditions for having complex solutions in a quadratic equation?
3. Can you solve the equation using a graphical method?
4. How does the discriminant determine the nature of solutions?
5. How would the solutions change if the discriminant were zero?

**Tip:** Always check the discriminant first to quickly determine the nature of the roots of a quadratic equation.

Understand how to determine the number and type of solutions for the quadratic equation 5x^2 + 5x - 3 = 0 using the discriminant formula. This problem involves calculating the discriminant, identifying the nature of roots, and applying quadratic equation principles. Suitable for Grades 10-12.

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