For what value of the constant k does the quadratic equation 0.5x^2 - 1.5x + k = 0 have tw… — ACT

For what value of the constant $k$ does the quadratic equation $0.5x^2 - 1.5x + k = 0$ have two complex roots of the form $a \pm 2i$ , where $a$ is a real number?

A
0.875
B
1.5625
C
2.125
3.125Answer
E
6.25

Answer

3.125

Applying the quadratic formula to

0.5x^2 - 1.5x + k = 0

yields

x = 1.5 \pm \sqrt{2.25 - 2k}

. For the roots to be of the form

a \pm 2i

, the imaginary component must be

2i

, which requires the radicand

2.25 - 2k

to equal

(2i)^2 = -4

. Solving the equation

2.25 - 2k = -4

yields

k = 3.125

. This corresponds to the option containing

3.125

Step-by-Step Solution

Identify the coefficients

a

b

, and

c

of the given quadratic equation

0.5x^2 - 1.5x + k = 0

a = 0.5

b = -1.5

, and

c = k

These coefficients are required to substitute into the quadratic formula.

Apply the quadratic formula,

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

, to express the roots of the equation.

x = \frac{-(-1.5) \pm \sqrt{(-1.5)^2 - 4(0.5)k}}{2(0.5)} = \frac{1.5 \pm \sqrt{2.25 - 2k}}{1} = 1.5 \pm \sqrt{2.25 - 2k}

Substituting the coefficients simplifies the expression for the roots of the quadratic equation.

Set the radical term equal to the imaginary component

2i

to find the value of

k

that yields complex roots of the form

a \pm 2i

\sqrt{2.25 - 2k} = 2i \Rightarrow 2.25 - 2k = (2i)^2 \Rightarrow 2.25 - 2k = -4

For the roots to have an imaginary part of

2i

, the term under the square root must equal

(2i)^2

, which is

-4

Solve the linear equation

2.25 - 2k = -4

for

k

-2k = -4 - 2.25 \Rightarrow -2k = -6.25 \Rightarrow k = 3.125

Isolating

k

gives the constant value that satisfies the root condition.

Key Concept

Using the quadratic formula and discriminant to determine the value of a parameter for a specified type of complex roots.

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