Roots of Quadratic Equation

CAT 1997 Slot 1 · QA · Medium · Algebra

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If the roots x1x_1 and x2x_2 of the quadratic equation x2−2x+c=0x^2 - 2x + c = 0 also satisfy the equation 7x2−4x1=477x_2 - 4x_1 = 47, then which of the following is true?

  1. A.

    c = -15

  2. B.

    x_1 = -5, x_2 = 3

  3. C.

    x_1 = 4.5, x_2 = -2.5

  4. D.

    None of these

Answer

A

Explanation

From the quadratic equation x2−2x+c=0x^2 - 2x + c = 0, sum of roots is: x1+x2=2  ⟹  x1=2−x2x_1 + x_2 = 2 \implies x_1 = 2 - x_2

Substitute x1=2−x2x_1 = 2 - x_2 into 7x2−4x1=477x_2 - 4x_1 = 47: 7x2−4(2−x2)=477x_2 - 4(2 - x_2) = 47 7x2−8+4x2=477x_2 - 8 + 4x_2 = 47 11x2=55  ⟹  x2=511x_2 = 55 \implies x_2 = 5

Then x1=2−5=−3x_1 = 2 - 5 = -3.

Product of roots c=x1x2=(−3)×5=−15c = x_1 x_2 = (-3) \times 5 = -15. Hence, c=−15c = -15 is true.

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