use the discriminant to determine whether the following equation have solutions that are : two different rational solutions : two different irrational solutions : exactly one rational solution : or two different imaginary solutions

7+5z^2=-8z

rearrange to the more usual form:

5z^2 + 8z + 7 = 0
the discriminant is 8^2-4*5*7 < 0

So, what does that tell you about the roots?

To determine the kind of solutions the equation 7+5z^2=-8z has, we need to use the discriminant. The discriminant is a mathematical formula that helps us determine the nature of solutions for a quadratic equation.

Let's start by writing the equation in standard quadratic form: 5z^2 + 8z + 7 = 0. Here, we have the quadratic equation in the form ax^2 + bx + c = 0, with a=5, b=8, and c=7.

The discriminant formula is given by Δ = b^2 - 4ac, where Δ represents the discriminant, b is the coefficient of the linear term, a is the coefficient of the quadratic term, and c is the constant term.

Plugging in the values from our equation, we get: Δ = (8)^2 - 4(5)(7).

Now, we can calculate: Δ = 64 - 140 = -76.

The discriminant Δ is -76, which is negative.

Based on the value of the discriminant, we can determine the nature of the solutions:

1. If Δ is positive, then we have two different real solutions.
2. If Δ is zero, then we have exactly one real solution (which is rational).
3. If Δ is negative, then we have two complex (imaginary) solutions.
4. If the coefficients in the equation lead to a perfect square trinomial, we have two different rational solutions.
5. If the coefficients do not lead to a perfect square trinomial, we have two different irrational solutions.

In the case of the equation 7+5z^2=-8z, since the discriminant Δ is negative (-76), it means that we have two different imaginary solutions.

Therefore, the equation 7+5z^2=-8z has two different imaginary solutions.