lf $\alpha,\ \beta$ are the roots of $4{x}^{2}+7{x}+2=0$, then the equation whose roots are $\alpha^{2},\ \beta^{2}$ is
- $16{x}^{2}-33{x}+4=0$
- $16{x}^{2}+33{x}+4=0$
- $4{x}^{2}-49{x}+2=0$
- $16{x}^{2}-4{x}+2=0$
If alpha and beta are roots of 4x^2 + 7x + 2 = 0, then alpha + beta = -7/4 and alpha*beta = 2/4 = 1/2. The new equation with roots alpha^2 and beta^2 is x^2 - (alpha^2 + beta^2)x + (alpha*beta)^2 = 0. Since alpha^2 + beta^2 = (alpha + beta)^2 - 2(alpha*beta) = 49/16 - 1 = 33/16, the equation is x^2 - (33/16)x + 1/4 = 0, which simplifies to 16x^2 - 33x + 4 = 0.
From the given quadratic equation 4x^2 + 7x + 2 = 0, Vieta's formulas tell us the sum of the roots is alpha + beta = -7/4 and the product is alpha * beta = 1/2. For the new equation with roots alpha^2 and beta^2, the sum of the new roots is alpha^2 + beta^2, which equals (alpha + beta)^2 - 2(alpha * beta). Substituting the values gives (-7/4)^2 - 2(1/2) = 49/16 - 1 = 33/16. The product of the new roots is (alpha * beta)^2 = (1/2)^2 = 1/4. Forming the new quadratic equation x^2 - (sum)x + (product) = 0 gives x^2 - (33/16)x + 1/4 = 0, and multiplying by 16 results in 16x^2 - 33x + 4 = 0.