Question

If $a$ and $b$ are the roots of the equation $2x^2 + 5x + 30 = 0$, find the value of $a^2 + b^2$.

• A. \(\frac{13}{4}\)
• B. \(-\frac{95}{4}\)
• C. \(\frac{25}{4}\)
• D. \(\frac{49}{4}\)

Hint:

For a quadratic equation, use the relationships between the sum and product of the roots to solve for unknown variables.

Answer 1

The Correct Option is B

Let \( a \) and \( b \) be the roots of the quadratic equation \( 2x^2 + 5x + 30 = 0 \).

By Vieta's formulas, the sum of the roots is

\[ a + b = -\frac{5}{2} \]

and the product of the roots is

\[ ab = \frac{30}{2} = 15. \]

We want to find the value of \( a^2 + b^2 \). We know that

\[ (a + b)^2 = a^2 + 2ab + b^2. \]

Then we have

\[ a^2 + b^2 = (a + b)^2 - 2ab. \]

Substituting the values we found earlier, we have

\[ a^2 + b^2 = \left( -\frac{5}{2} \right)^2 - 2(15) = \frac{25}{4} - 30 = \frac{25}{4} - \frac{120}{4} = \frac{25 - 120}{4} = \frac{-95}{4}. \]

Therefore, \( a^2 + b^2 = -\frac{95}{4} \).

Final Answer: The final answer is \( \boxed{-\frac{95}{4}} \)