Multiple choice

If the roots of the quadratic equation 2x2 – (7m – 2)x + 18 = 0 are equal and m > 0, then find the quadratic equation if one of the roots is 'm' and the other is -3.

  1. x2 – 9 = 0

  2. x2 + x + 6 = 0

  3. x2 + x - 6 = 0

  4. More than one of the above
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C Correct answer
Explanation

For equal roots, the discriminant D = b^2 - 4ac must be 0. Solving (7m - 2)^2 - 4(2)(18) = 0 gives (7m - 2)^2 = 144, so 7m - 2 = 12 (since m > 0), resulting in m = 2. With roots 2 and -3, the quadratic equation is (x - 2)(x + 3) = x^2 + x - 6 = 0.

AI explanation

For the equation 2x^2 - (7m - 2)x + 18 = 0 to have equal roots, its discriminant must be zero, so (7m - 2)^2 - 4(2)(18) = 0. This simplifies to (7m - 2)^2 = 144, yielding 7m - 2 = 12 or 7m - 2 = -12; solving these gives m = 2 or m = -10/7. Since m > 0, we take m = 2. The required new equation has roots 2 and -3, so the sum of the roots is -1 and the product is -6, which leads to the equation x^2 + x - 6 = 0.

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