Multiple choice

lf the roots of equation ${ x }^{ 2 }+bx+ac=0$ are $\alpha $ and $\beta$, the roots of the equation ${ x }^{ 2 }+ax+bc=0$ are $\alpha $ and $\gamma $, then the values of $\alpha,\ \beta,\ \gamma$ are respectively

  1. ${c},{b}, a$
  2. ${c}, {a}, {b}$
  3. ${a}, {c}, {b}$
  4. ${b}, {a}, {c}$
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B Correct answer
Explanation

For x^2 + bx + ac = 0, roots are alpha and beta. For x^2 + ax + bc = 0, roots are alpha and gamma. Subtracting equations: (b-a)x + (ac-bc) = 0, so (b-a)x = c(b-a). Thus x = c. Since alpha is a root of both, alpha = c. Substituting alpha = c into the first equation: c^2 + bc + ac = 0, c(c+b+a) = 0. This implies alpha=c, beta=a, gamma=b.

AI explanation

Since alpha is a root of both x^2 + bx + ac = 0 and x^2 + ax + bc = 0, subtracting the two equations yields (b - a)x + c(a - b) = 0. Factoring this gives (b - a)(x - c) = 0, so alpha = c. Substituting x = c into the first equation gives c^2 + bc + ac = 0, which means c + a + b = 0 and b = -(a + c). Using the sum of roots for the first equation, alpha + beta = -b, so c + beta = a + c, giving beta = a. Using the sum of roots for the second equation, alpha + gamma = -a, so c + gamma = -a, giving gamma = -a - c which equals b. Therefore, alpha, beta, and gamma are c, a, and b respectively.

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