Multiple choice

Let us consider a quadratic equation $x^2+3ax+2a^2=0$ If this equation has roots $\alpha ,\beta $ and it is given that $\alpha^2 +\beta ^2=5$, then v alue of discriminant, $D$, for the above quadratic equation is

  1. $D>0$
  2. $D<0$
  3. $D=0$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For the roots, alpha^2 + beta^2 = (alpha + beta)^2 - 2alpha beta = 9a^2 - 4a^2 = 5a^2. Since this equals 5, a^2 = 1. The discriminant is 9a^2 - 8a^2 = a^2 = 1, so D > 0.

AI explanation

Using Vieta's formulas for x^2 + 3ax + 2a^2 = 0, the sum of the roots is alpha + beta = -3a and the product is alpha*beta = 2a^2. Apply the algebraic identity alpha^2 + beta^2 = (alpha + beta)^2 - 2*alpha*beta to get (-3a)^2 - 2(2a^2) = 5. This simplifies to 9a^2 - 4a^2 = 5, so 5a^2 = 5 and a^2 = 1. The discriminant of the quadratic equation is D = (3a)^2 - 4(1)(2a^2) = 9a^2 - 8a^2 = a^2. Substituting a^2 = 1 gives D = 1, meaning D > 0.

Keep practicing — related questions