Question:
If $a$ and $b$ are the roots of the equation $2x^2 + 5x + 30 = 0$, find the value of $a^2 + b^2$.
If $a$ and $b$ are the roots of the equation $2x^2 + 5x + 30 = 0$, find the value of $a^2 + b^2$.
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For a quadratic equation, use the relationships between the sum and product of the roots to solve for unknown variables.
Updated On: May 24, 2025
- \(\frac{13}{4}\)
- \(-\frac{95}{4}\)
- \(\frac{25}{4}\)
- \(\frac{49}{4}\)
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The Correct Option is B
Solution and Explanation
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 \( \boxed{-\frac{95}{4}} \)
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