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.
-$- 5 - 5 a ^ { 2 } = - 7 a - 7$
A) Two distinct nonreal complex solutions
B) Two distinct rational solutions
C) One rational solution (a double solution)
D) Two distinct irrational solutions

Question

Quizplus · Accepted Answer

The given equation is $-5 - 5a^2 = -7a - 7$. First, we rearrange it into standard quadratic form $ax^2 + bx + c = 0$, which gives us $5a^2 + 7a + 2 = 0$. The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by $D = b^2 - 4ac$. Substituting the values from our equation, we get $D = (7)^2 - 4(5)(2) = 49 - 40 = 9$. Since the discriminant is positive and a perfect square, the equation has two distinct rational solutions. However, the correct explanation leads to the conclusion that there was an error in my initial assessment. The correct answer should be that the equation has two distinct rational solutions, not irrational as initially stated. This mistake was due to an incorrect interpretation of the discriminant's implications. The correct choice should have been B, indicating two distinct rational solutions, as a positive and perfect square discriminant indicates rational solutions.

Evaluate the Discriminant for the Equation −5−5a2=−7a−7- 5 - 5 a ^ { 2 } = - 7 a - 7−5−5a2=−7a−7

Evaluate the Discriminant for the Equation $- 5 - 5 a ^ { 2 } = - 7 a - 7$