Use the discriminant to determine whether the quadratic equation has two unequal real solutions, a repeated real solution, or no real solution, without solving the equation.
5x^2-2x+6=0
A. Two Unequal Solutions
B. One repeated Solution
C. No real solutions
I chose C is that correct?
correct, the discriminant has a negative value
To determine the type of solutions without solving the quadratic equation, you can use the discriminant. The discriminant is the expression underneath the square root sign in the quadratic formula and is calculated as follows:
Discriminant (D) = b^2 - 4ac
In the given quadratic equation 5x^2 - 2x + 6 = 0, we can identify the values of a, b, and c from the equation as follows:
a = 5
b = -2
c = 6
Now, substitute these values into the discriminant formula:
D = (-2)^2 - 4(5)(6)
D = 4 - 120
D = -116
By calculating the discriminant, we find that it equals -116.
Based on the value of the discriminant, we can determine the type of solutions:
If the discriminant (D) is greater than 0, the quadratic equation has two unequal real solutions.
If the discriminant (D) is equal to 0, the quadratic equation has one repeated real solution.
If the discriminant (D) is less than 0, the quadratic equation has no real solutions.
In this case, since the value of the discriminant is -116 (which is less than 0), it means that the quadratic equation 5x^2 - 2x + 6 = 0 has no real solutions.
Therefore, selecting option C, "No real solutions," is indeed correct.