QuestionAugust 10, 2025

How many solutions does the equation x^2+(1)/(2)x+5=0 have? (A) No real solution (B) 1 real solution (C) 2 rational solutions (D) 2 irrational solutions (E) Cannot be determined

How many solutions does the equation x^2+(1)/(2)x+5=0 have? (A) No real solution (B) 1 real solution (C) 2 rational solutions (D) 2 irrational solutions (E) Cannot be determined
How many solutions does the equation x^2+(1)/(2)x+5=0 have?
(A) No real solution
(B) 1 real solution
(C) 2 rational solutions
(D) 2 irrational solutions
(E) Cannot be determined

Solution
4.2(248 votes)

Answer

(A) No real solution Explanation 1. Calculate the discriminant Use the formula for the discriminant \Delta = b^2 - 4ac. Here, a = 1, b = \frac{1}{2}, and c = 5. Calculate \Delta = \left(\frac{1}{2}\right)^2 - 4 \cdot 1 \cdot 5 = \frac{1}{4} - 20 = -\frac{79}{4}. 2. Determine the number of solutions Since \Delta < 0, the equation has no real solutions.

Explanation

1. Calculate the discriminant<br /> Use the formula for the discriminant $\Delta = b^2 - 4ac$. Here, $a = 1$, $b = \frac{1}{2}$, and $c = 5$. Calculate $\Delta = \left(\frac{1}{2}\right)^2 - 4 \cdot 1 \cdot 5 = \frac{1}{4} - 20 = -\frac{79}{4}$.<br />2. Determine the number of solutions<br /> Since $\Delta < 0$, the equation has no real solutions.
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