The quadratic equation x^2 - kx + 36 = 0, where k is a positive constant, has two distinct… — GRE General Test

The quadratic equation $x^2 - kx + 36 = 0$ , where $k$ is a positive constant, has two distinct real roots $r_1$ and $r_2$ such that $r_2 - r_1 = 5$ . What is the value of $k$ ?

Answer: 13

Answer

The value of

k

is 13.

According to Vieta's formulas, for the quadratic equation

x^2 - kx + 36 = 0

, the sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 r_2 = 36

. Using the identity

(r_2 - r_1)^2 = (r_1 + r_2)^2 - 4r_1 r_2

, we substitute the known values

r_2 - r_1 = 5

r_1 + r_2 = k

, and

r_1 r_2 = 36

. This gives

5^2 = k^2 - 4(36)

, which simplifies to

25 = k^2 - 144

. Solving for

k^2

gives

k^2 = 169

. Since

k

is specified as a positive constant,

k = 13

Step-by-Step Solution

Apply Vieta's formulas to the given quadratic equation

The sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 r_2 = 36

For any quadratic equation

ax^2 + bx + c = 0

, the sum of roots is

-b/a

and the product of roots is

c/a

Relate the difference of the roots to their sum and product

(r_2 - r_1)^2 = (r_1 + r_2)^2 - 4r_1 r_2

Expanding both sides shows that

r_2^2 - 2r_1 r_2 + r_1^2 = r_1^2 + 2r_1 r_2 + r_2^2 - 4r_1 r_2

, which is an algebraic identity.

Substitute the known values into the identity

5^2 = k^2 - 4(36) \implies 25 = k^2 - 144

We are given that

r_2 - r_1 = 5

r_1 r_2 = 36

, and

r_1 + r_2 = k

Solve for the positive constant

k

k^2 = 169 \implies k = 13

Adding 144 to both sides gives

k^2 = 169

. Taking the positive square root because

k > 0

yields

k = 13

Key Concept

Vieta's Formulas and Root Difference Identity

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