In the quadratic equation x^2 - 15x + k = 0, where k is a constant, one of the real soluti… — SAT

In the quadratic equation $x^2 - 15x + k = 0$ , where $k$ is a constant, one of the real solutions is $4$ times the other solution. What is the value of $k$ ?

Answer: 36

Answer

The correct answer is 36. By Vieta's formulas, the sum of the roots of the quadratic equation

x^2 - 15x + k = 0

15

, and the product of the roots is

k

. Let the roots be

r

and

4r

. Their sum is

r + 4r = 5r = 15

, which gives

r = 3

. The roots are

3

and

4(3) = 12

. The product of the roots is

3 \times 12 = 36

, which represents the constant term

k

Step-by-Step Solution

Express the relationship between the two roots.

Let the two roots be

r

and

4r

We are given that one of the real solutions is 4 times the other solution.

Use Vieta's formulas to set up an equation for the sum of the roots.

r + 4r = 15 \implies 5r = 15

For a quadratic equation in the form

x^2 - sx + p = 0

, the sum of the roots is equal to

s

Solve for the variable

r

and find both roots.

r = 3

, so the roots are

3

and

4(3) = 12

Dividing both sides of

5r = 15

by 5 gives

r = 3

. Substituting this back gives the two roots.

Calculate the product of the roots to find the value of

k

k = 3 \times 12 = 36

For a quadratic equation in the form

x^2 - sx + p = 0

, the product of the roots is equal to

p

, which in this case is

k

Key Concept

Using Vieta's formulas to find relationships between the coefficients and roots of a quadratic equation.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic