BTGmoderatorDC wrote:What are the roots of the quadratic equation x^2 + bx + c = 0 if the roots are distinct and at equal distance from 5 on the number line?

(1) The product of the roots of the equation x^2 + bx + c = 0 is 21

(2) x - 7 is a factor of the expression x^2 + bx + c

OA D

Source: e-GMAT

Say the two roots of the equation are p and q such that q > p. Note that p < 5 and q > 5; thus, p - 5 = q - 5 => p + q = 10.

We have to get the value of p and q.

Let's take each statement one by one.

(1) The product of the roots of the equation x^2 + bx + c = 0 is 21.

We know that for a quadratic equation ax^2 + bx = c = 0, the sum of roots = -b/a and the product of the roots = c/a

Thus, for the equation x^2 + bx + c = 0, the sum of roots = -b/a = -b/1 = 10 (given) => b = -10 and the product of the roots = c/a = c/1 = 21 => c = 21 (given)

Thus, the equation is x^2 - 10x + 21 = 0. We can factorize this equation and get the two roots. Sufficient. Let's do it for the completeness.

x^2 - 10x + 21 = 0 => x^2 - 7x -3x + 21 = 0 => x(x -7) - 3(x -7) = 0 => (x - 7)(x -3) = 0 => x = 3 or 7. Sufficient.

(2) (x - 7) is a factor of the expression x^2 + bx + c.

Given that (x - 7) is a factor of the expression x^2 + bx + c, we have x = 7 is a root of the equation x^2 + bx + c = 0.

Since 7 > 5, the other root would be less than 5. We have 7 - 5 = 2 = distance between one of the roots; thus, 5 - 2 = 3 = the second root. Sufficient.

The correct answer: D

Hope this helps!

-Jay
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