Evaluate the discriminant for the equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or nonreal complex.
-$2 - 5 a ^ { 2 } = 4 a - 3$
A) Two distinct nonreal complex solutions
B) One rational solution (a double solution)
C) Two distinct rational solutions
D) Two distinct irrational solutions

Question

Quizplus · Accepted Answer

The discriminant of a quadratic equation in the form $ax^2 + bx + c = 0$ is given by $b^2 - 4ac$. First, we need to rearrange the given equation into this standard form: $2 - 5a^2 = 4a - 3$, which simplifies to $5a^2 + 4a - 5 = 0$. Here, $a = 5$, $b = 4$, and $c = -5$. The discriminant is $b^2 - 4ac = 4^2 - 4(5)(-5) = 16 + 100 = 116$. Since the discriminant is positive and not a perfect square, the equation has two distinct irrational solutions.

Evaluate the Discriminant for the Equation 2−5a2=4a−32 - 5 a ^ { 2 } = 4 a - 32−5a2=4a−3

Evaluate the Discriminant for the Equation $2 - 5 a ^ { 2 } = 4 a - 3$