In the quadratic equation x^2 - kx + 36 = 0, k is a positive constant. If the difference b… — SAT

In the quadratic equation $x^2 - kx + 36 = 0$ , $k$ is a positive constant. If the difference between the two solutions to the equation is $5$ , what is the value of $k$ ?

Answer: 13

Answer

The value of the positive constant

k

13

For the quadratic equation

x^2 - kx + 36 = 0

with solutions

r_1

and

r_2

, the sum of the solutions is

r_1 + r_2 = k

and the product of the solutions is

r_1 \cdot r_2 = 36

. Given that the difference between the two solutions is

5

, we can write

|r_1 - r_2| = 5

. Squaring both sides yields

(r_1 - r_2)^2 = 25

. Using the algebraic identity

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

, we substitute the known values to obtain

25 = k^2 - 4(36)

, which simplifies to

25 = k^2 - 144

. Solving for

k^2

gives

k^2 = 169

. Since

k

is positive,

k = 13

. Alternatively, we can find two numbers whose product is

36

and whose difference is

5

. These numbers are

9

and

4

, because

9 \cdot 4 = 36

and

9 - 4 = 5

. The sum of these solutions is

9 + 4 = 13

, which matches the coefficient of the linear term,

k

Step-by-Step Solution

Relate the roots of the quadratic equation

x^2 - kx + 36 = 0

to its coefficients using Vieta's formulas.

The sum of the roots is

r_1 + r_2 = k

and the product of the roots is

r_1 \cdot r_2 = 36

This sets up the system of equations representing the roots.

Express the given root difference of

5

mathematically and square it.

|r_1 - r_2| = 5 \implies (r_1 - r_2)^2 = 25

Squaring the difference allows us to use standard algebraic identities.

Apply the identity

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

to substitute the sum and product expressions.

25 = k^2 - 4(36)

This converts the relationship between the roots into an equation with the single variable

k

Solve the equation for

k

, selecting the positive value.

25 = k^2 - 144 \implies k^2 = 169 \implies k = 13

(since

k > 0

To determine the final value of

k

satisfying the constraint that

k

is a positive constant.

Key Concept

Relationship between the roots and coefficients of a quadratic equation (Vieta's formulas)

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