If the roots of the quadratic equation x^2 - (k - 2)x + (k + 1) = 0 have a sum that is equ… — GMAT

If the roots of the quadratic equation $x^2 - (k - 2)x + (k + 1) = 0$ have a sum that is equal to twice their product, what is the value of $k$ ?

Answer: -4

The value of

k

-4

By Vieta's formulas, the sum of the roots of

x^2 - (k-2)x + (k+1) = 0

k-2

and the product of the roots is

k+1

. Equating the sum to twice the product gives

k - 2 = 2(k + 1)

, which simplifies to

k - 2 = 2k + 2

and yields

k = -4

. Evaluating the discriminant with

k = -4

gives

\Delta = (-6)^2 - 4(-3) = 48 > 0

, ensuring real roots exist.

Apply Vieta's formulas to express the sum and product of the roots in terms of

k

Sum of roots

= -\frac{b}{a} = k - 2

, and product of roots

= \frac{c}{a} = k + 1

For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

Set up an equation based on the condition that the sum of the roots equals twice their product.

k - 2 = 2(k + 1)

This directly translates the verbal condition given in the problem.

Solve the linear equation for

k

k - 2 = 2k + 2 \implies -4 = k

Subtracting

k

and 2 from both sides isolates

k

Check the discriminant to ensure real roots exist when

k = -4

\Delta = b^2 - 4ac = (-6)^2 - 4(1)(-3) = 36 + 12 = 48 > 0

A positive discriminant guarantees that the quadratic equation has two distinct real roots.

Vieta's Formulas for Quadratic Equations

Question