What are the solutions for x in the quadratic equation 0.2x^2 - 0.6x + 2.2 = 0? — ACT

What are the solutions for $x$ in the quadratic equation $0.2x^2 - 0.6x + 2.2 = 0$ ?

The correct answer is

\frac{3 \pm i\sqrt{35}}{2}

The correct answer is the pair of complex solutions

\frac{3 \pm i\sqrt{35}}{2}

. By multiplying the equation

0.2x^2 - 0.6x + 2.2 = 0

5

, we get the equivalent equation with integer coefficients,

x^2 - 3x + 11 = 0

. Applying the quadratic formula with

a = 1

b = -3

, and

c = 11

yields a discriminant of

9 - 44 = -35

. Since the discriminant is negative, the solutions are complex:

\frac{3 \pm i\sqrt{35}}{2}

Multiply both sides of the quadratic equation by

5

to eliminate the decimal coefficients.

The equation

0.2x^2 - 0.6x + 2.2 = 0

becomes

x^2 - 3x + 11 = 0

Working with integer coefficients simplifies algebraic manipulation and reduces the risk of calculation errors.

Identify the coefficients

a

b

, and

c

in the standard quadratic form

ax^2 + bx + c = 0

a = 1

b = -3

, and

c = 11

These values are required to apply the quadratic formula.

Substitute the coefficients into the quadratic formula,

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

x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(11)}}{2(1)} = \frac{3 \pm \sqrt{9 - 44}}{2} = \frac{3 \pm \sqrt{-35}}{2}

The quadratic formula provides the exact solutions for any quadratic equation.

Simplify the radical using the imaginary unit,

i = \sqrt{-1}

\sqrt{-35} = \sqrt{35} \cdot \sqrt{-1} = i\sqrt{35}

, so the solutions are

x = \frac{3 \pm i\sqrt{35}}{2}

Standard mathematical notation represents the square root of a negative number using

i

Using the quadratic formula to solve quadratic equations with decimal coefficients and complex roots.

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