For a certain quadratic equation 0.25x^2 - kx + 4.5 = 0, where k is a constant, the differ… — ACT

For a certain quadratic equation $0.25x^2 - kx + 4.5 = 0$ , where $k$ is a constant, the difference between the two real solutions is exactly $3$ . What is the positive value of $k$ ?

Answer: 2.25

Answer

The positive value of

k

2.25

By applying the root difference formula

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

with

a = 0.25

b = -k

, and

c = 4.5

, we get the equation

\frac{\sqrt{k^2 - 4.5}}{0.25} = 3

. This simplifies to

4\sqrt{k^2 - 4.5} = 3

. Dividing by

4

and squaring both sides gives

k^2 - 4.5 = 0.5625

, which means

k^2 = 5.0625

. Taking the positive square root yields

k = 2.25

Step-by-Step Solution

Identify the coefficients of the quadratic equation and express the formula for the difference of the roots.

The coefficients are

a = 0.25

b = -k

, and

c = 4.5

. The difference between the roots is

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

This sets up the algebraic relationship between the root difference and the coefficients of the quadratic equation.

Substitute the coefficients and the given root difference of

3

into the difference formula.

\frac{\sqrt{(-k)^2 - 4(0.25)(4.5)}}{0.25} = 3

This creates an equation in terms of the variable

k

using the given problem constraints.

Simplify the equation and isolate the radical term.

4\sqrt{k^2 - 4.5} = 3 \implies \sqrt{k^2 - 4.5} = 0.75

Simplifying the fraction makes it easier to isolate the radical prior to squaring.

Square both sides of the equation to eliminate the radical and solve for

k^2

k^2 - 4.5 = 0.5625 \implies k^2 = 5.0625

Squaring is the inverse operation of the square root, allowing us to solve for

k^2

Calculate the positive square root of

5.0625

to find the value of

k

k = 2.25

Since the question asks for the positive value of

k

, we choose the positive square root.

Key Concept

Relating the difference of the roots of a quadratic equation to its coefficients using the discriminant and the quadratic formula.

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