Please Help!!
Use the discriminant to determine whether the follow equations have solutions that are:
-two different rational solutions
-two different irrational solutions
-exactly one rational solution or
-two different imaginary solutions.
8x^2+7x+3=0
evaluate b^2 - 4ac
= 49-4(8)(3) = -47
look in your notes or textbook
If the discriminant is negative, then there are two different imaginary roots or solutions
To determine the nature of the solutions for the given quadratic equation, we need to calculate the discriminant. The discriminant is a mathematical expression that helps us determine whether the solutions are rational, irrational, real, imaginary, or unique.
The general form of a quadratic equation is given by ax^2 + bx + c = 0, where a, b, and c are coefficients.
In this case, the equation is 8x^2 + 7x + 3 = 0, so our coefficients are:
a = 8
b = 7
c = 3
The discriminant (denoted as Δ) for a quadratic equation is calculated using the formula:
Δ = b^2 - 4ac
Let's substitute the values into the formula and solve for Δ:
Δ = (7)^2 - 4(8)(3)
Δ = 49 - 96
Δ = -47
Now that we have calculated the discriminant, we can analyze its value to determine the nature of the solutions:
1. If the discriminant (Δ) is positive and a perfect square, the equation has two different rational solutions.
2. If the discriminant (Δ) is positive but not a perfect square, the equation has two different irrational solutions.
3. If the discriminant (Δ) is zero, the equation has exactly one rational solution.
4. If the discriminant (Δ) is negative, the equation has two different imaginary solutions.
In our case, the discriminant is negative (Δ = -47), which means the equation has two different imaginary solutions.
Therefore, the quadratic equation 8x^2 + 7x + 3 = 0 has two different imaginary solutions.