use thr discriminant to determine wheather the following equations have solutions that are two different rational solutions;two different irrational solutions; or two imaginary solutions.8x^2+7x+3=0
8x^2 + 7x + 3 = 0.
D = B^2 - 4AC = 49 - 96 = -47.
B^2 < 4AC,
Therefore, we have 2 imaginary solutions.
To determine the nature of the solutions for the quadratic equation 8x^2 + 7x + 3 = 0, we need to use the discriminant. The discriminant, denoted as ∆, is found using the formula:
∆ = b^2 - 4ac
In the given equation, the coefficients are:
a = 8
b = 7
c = 3
Substituting these values into the formula, we find:
∆ = (7)^2 - 4(8)(3)
= 49 - 96
= -47
Now, based on the value of the discriminant, we can determine the nature of the solutions:
1. If the discriminant (∆) is positive (∆ > 0), then there are two different real solutions (either rational or irrational).
2. If the discriminant (∆) is zero (∆ = 0), then there is one real solution (rational or irrational), which is known as a "repeated" or "double" solution.
3. If the discriminant (∆) is negative (∆ < 0), then there are two imaginary solutions.
In our case, the discriminant is ∆ = -47, which is negative (∆ < 0). Therefore, the equation 8x^2 + 7x + 3 = 0 has two imaginary solutions.