Use the discriminant to determine the number and type of solutions for the given equation. Either A, B, C, OR D
2x^2 = 7x - 8
A One (repeated) rational solution
B Two irrational solutions
C Two imaginary solutions
D Two rational solutions
2x^2-7x+8=0
b^2-4ac= 49-64=-15 So that puts A, D as not the answers.
To use the discriminant to determine the number and type of solutions for the given equation, we need to write the equation in the standard form, which is of the form ax^2 + bx + c = 0.
Given equation: 2x^2 = 7x - 8
First, rewrite the equation in the standard form by moving all terms to one side:
2x^2 - 7x + 8 = 0
Now we can identify the values of a, b, and c:
a = 2
b = -7
c = 8
The discriminant (denoted by Δ) is given by the formula: Δ = b^2 - 4ac
Substitute the values of a, b, and c into the formula:
Δ = (-7)^2 - 4(2)(8)
= 49 - 64
= -15
The value of the discriminant is -15.
Now, let's analyze the value of the discriminant:
1. If the discriminant is positive (Δ > 0), then the equation has two distinct real solutions.
2. If the discriminant is zero (Δ = 0), then the equation has one (repeated) real solution.
3. If the discriminant is negative (Δ < 0), then the equation has two complex solutions (imaginary solutions).
In this case, since the value of the discriminant is negative (Δ = -15), the equation has two complex solutions.
Therefore, the answer is:
C Two imaginary solutions
To determine the number and type of solutions for the equation 2x^2 = 7x - 8, we need to calculate the discriminant.
The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. In this case, a = 2, b = -7, and c = -8.
The discriminant (D) of a quadratic equation is given by the formula D = b^2 - 4ac.
Substituting the values, we have D = (-7)^2 - 4(2)(-8) = 49 + 64 = 113.
Now let's interpret the discriminant:
1. If the discriminant (D) is greater than 0, then the equation has two distinct real solutions. (Two rational solutions)
2. If the discriminant (D) is equal to 0, then the equation has one repeated real solution. (One repeated rational solution)
3. If the discriminant (D) is less than 0, then the equation has two complex solutions. (Two imaginary solutions)
In this case, the discriminant is 113 which is greater than 0. Therefore, the equation 2x^2 = 7x - 8 has two distinct real solutions.
So the answer is D. Two rational solutions.