The equation whose roots are the squares of the roots of the equation $2x^2 + 3x + 1 = 0$ is
- $4x^2 + 5x + 1 = 0$
- $4x^2 - x + 1 = 0$
- $4x^2 - 5x - 1 = 0$
- $4x^2 - 5x + 1 = 0$
- $4x^2 + 5x - 1 = 0$
Let roots of 2x^2 + 3x + 1 = 0 be a, b. a+b = -3/2, ab = 1/2. We want an equation with roots a^2, b^2. Sum = a^2+b^2 = (a+b)^2 - 2ab = 9/4 - 1 = 5/4. Product = a^2b^2 = (ab)^2 = 1/4. Equation: x^2 - (5/4)x + 1/4 = 0, or 4x^2 - 5x + 1 = 0.
Using the sum and product of roots formula for 2x^2 + 3x + 1 = 0, the sum of the roots (alpha + beta) is -3/2 and the product (alpha * beta) is 1/2. For the new equation, the sum of the roots will be alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha * beta) = (-3/2)^2 - 2(1/2) = 9/4 - 1 = 5/4. The product of the new roots is alpha^2 * beta^2 = (1/2)^2 = 1/4. Forming the new quadratic equation x^2 - (sum)x + product = 0 gives x^2 - 5/4x + 1/4 = 0, and multiplying by 4 gives 4x^2 - 5x + 1 = 0.