Multiple choice

If $\displaystyle \alpha ,\beta $ are the roots of a quadratic equation $\displaystyle { x }^{ 2 }-3kx+{ k }^{ 2 }=0$, find the values of k. If $\displaystyle { \alpha }^{ 2 }+{ \beta }^{ 2 }=\frac { 7 }{ 4 } $.

  1. $\displaystyle \pm \frac { 1 }{ 3 } $
  2. $\displaystyle \pm \frac { 1 }{ 2 } $
  3. $\displaystyle \pm \frac { 1 }{ 4 } $
  4. $\displaystyle \pm \frac { 1 }{ 5 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For x^2 - 3kx + k^2 = 0, alpha+beta = 3k and alpha*beta = k^2. alpha^2+beta^2 = (alpha+beta)^2 - 2*alpha*beta = (3k)^2 - 2(k^2) = 9k^2 - 2k^2 = 7k^2. Given 7k^2 = 7/4, so k^2 = 1/4, k = +/- 1/2.

AI explanation

For the equation x^2 - 3kx + k^2 = 0, the sum of the roots alpha and beta is 3k, and their product is k^2. Using the sum of squares identity, alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha)(beta), we substitute the given value to get 7/4 = (3k)^2 - 2(k^2). This simplifies to 7/4 = 9k^2 - 2k^2, resulting in 7/4 = 7k^2. Dividing both sides by 7 gives k^2 = 1/4, which means k = +1/2 or -1/2.

Keep practicing — related questions