Multiple choice

Ruhi and Tara were asked to solve a quadratic equation $ax^{2} + bx + c = 0, a\neq 0$$. Ruhi made some mistake in writing b and found the roots as $3$ and $-\dfrac {1}{2}$. Tara, too, made mistake in writing c and found the roots $-1$ and $-\dfrac {1}{4}$. Find the correct roots of the given equation

  1. $-2, \dfrac {3}{4}$
  2. $3, -1$
  3. $-\dfrac {1}{2}, -1$
  4. $3, -\dfrac {1}{4}$
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A Correct answer
Explanation

Ruhi's roots 3 and -1/2 imply the sum is 5/2, but she erred in b, so the product c/a = 3 * (-1/2) = -3/2 is correct. Tara's roots -1 and -1/4 imply the product 1/4 is wrong, but the sum -b/a = -1 + (-1/4) = -5/4 is correct. Thus, -b/a = -5/4 and c/a = -3/2. The equation is x^2 + (5/4)x - 3/2 = 0, or 4x^2 + 5x - 6 = 0. Factoring gives (4x-3)(x+2) = 0, so roots are 3/4 and -2.

AI explanation

Ruhi found the correct value for c by using her roots in Vieta's formulas, giving c/a = (3) * (-1/2) = -3/2. Tara found the correct value for b by using her roots, giving -b/a = (-1) + (-1/4) = -5/4, so b/a = 5/4. For the correct equation ax^2 + bx + c = 0, divide by a to get x^2 + (5/4)x - 3/2 = 0. Multiplying the entire equation by 4 yields 4x^2 + 5x - 6 = 0, which factors into (4x-3)(x+2) = 0. The correct roots are -2 and 3/4.

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