One of the solutions to the quadratic equation 2x^2 - kx + 18 = 0 is exactly four times th… — ACT

One of the solutions to the quadratic equation $2x^2 - kx + 18 = 0$ is exactly four times the other solution. If $k$ is a positive constant, what is the value of $k$ ?

Answer: 15

Answer

By representing the roots as

r

and

4r

, we can use Vieta's formulas to find that the product of the roots is

4r^2 = 9

, which gives

r = 1.5

. The sum of the roots is

5r = k/2

, which gives

k = 10(1.5) = 15

Step-by-Step Solution

Represent the roots of the quadratic equation.

Let the two solutions be

r

and

4r

, where

r

is a real number.

We are given that one solution is exactly four times the other.

Apply Vieta's formulas for the product of the roots.

r \cdot 4r = 4r^2 = \frac{18}{2} = 9

For a quadratic equation

ax^2 + bx + c = 0

, the product of the roots is equal to

\frac{c}{a}

Solve for the root

r

r^2 = \frac{9}{4} \Rightarrow r = 1.5

(since

k

is positive,

r

must be positive)

Solving the equation

4r^2 = 9

gives

r = \pm 1.5

. Since the sum of the roots is positive, we select the positive root.

Apply Vieta's formulas for the sum of the roots to find

k

r + 4r = 5r = -\frac{-k}{2} = \frac{k}{2} \Rightarrow k = 10r = 10(1.5) = 15

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is equal to

-\frac{b}{a}

Key Concept

Using Vieta's formulas to relate the roots of a quadratic equation to its coefficients.

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