Multiple choice

For what value of $a$ the difference of the roots of the equation $2x^2 - (a + 1)x + (a - 1) = 0$ is equal to their product?

  1. $0$
  2. $1$
  3. $-1$
  4. $2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Roots r1, r2. r1-r2 = r1*r2. (r1-r2)^2 = (r1+r2)^2 - 4r1r2. So (r1*r2)^2 = (r1+r2)^2 - 4r1r2. Sum = (a+1)/2, Product = (a-1)/2. ((a-1)/2)^2 = ((a+1)/2)^2 - 4((a-1)/2). (a-1)^2/4 = (a+1)^2/4 - 2(a-1). (a^2-2a+1) = (a^2+2a+1) - 8a + 8. -2a = -6a + 8. 4a = 8, a=2.

AI explanation

For the quadratic 2x squared minus (a plus 1)x plus (a minus 1) equals 0, the sum of the roots is (a plus 1) divided by 2 and the product is (a minus 1) divided by 2. The difference of the roots squared equals (sum squared minus 4 times product), which gives the equation ((a plus 1) divided by 2) squared minus 4 times ((a minus 1) divided by 2) equaling ((a minus 1) divided by 2) squared. Multiplying by 4 and expanding both sides gives a squared plus 2a plus 1 minus 8a plus 8 equals a squared minus 2a plus 1. Solving the resulting linear equation minus 8a plus 8 equals minus 2a yields a equals 2.

Keep practicing — related questions