Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $2x^2-3x-5=0$, Find $\alpha^2+\beta^2$

  1. $\dfrac 94$
  2. $\dfrac {29}4$
  3. $\dfrac 46$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For 2x^2 - 3x - 5 = 0, alpha + beta = 3/2 and alpha * beta = -5/2. alpha^2 + beta^2 = (alpha + beta)^2 - 2 * alpha * beta = (3/2)^2 - 2 * (-5/2) = 9/4 + 5 = 29/4.

AI explanation

By Vieta's formulas for the equation 2x squared minus 3x minus 5 equals 0, the sum of the roots alpha plus beta is negative b over a, which is 3 over 2. We use the algebraic identity alpha squared plus beta squared equals (alpha plus beta) squared minus 2 alpha beta, where the product alpha beta is c over a, equaling negative 5 over 2. Substituting these values gives (3 over 2) squared minus 2 times (negative 5 over 2), which calculates to 9 over 4 plus 5, resulting in 29 over 4.

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