Multiple choice

Find the value of a for which the sum of the squares of the roots of the equation $x^2 - (a - 2)x - a- 1 = 0$ assumes the least value.

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

The sum of the squares of the roots of x^2 - (a-2)x - (a+1) = 0 is (sum of roots)^2 - 2(product of roots). This is (a-2)^2 - 2(-(a+1)) = a^2 - 4a + 4 + 2a + 2 = a^2 - 2a + 6. This quadratic in 'a' reaches its minimum at a = -(-2)/(2*1) = 1.

AI explanation

For the given equation x^2 - (a - 2)x - (a + 1) = 0, the sum of the roots is (a - 2) and the product of the roots is -(a + 1). The sum of the squares of the roots can be written as (Sum)^2 - 2(Product). Substituting the Vieta's formulas gives (a - 2)^2 - 2(-a - 1), which simplifies to a^2 - 4a + 4 + 2a + 2 = a^2 - 2a + 6. To find the minimum of this quadratic expression, we find the vertex at a = -(-2)/2(1) = 1. Therefore, the required value is a = 1.

Keep practicing — related questions