What are the solutions to the quadratic equation x^2 - \frac{5}{2}x + 1 = 0? — ACT

What are the solutions to the quadratic equation $x^2 - \frac{5}{2}x + 1 = 0$ ?

The solutions are

x = \frac{1}{2}

and

x = 2

The correct answer provides the exact solutions to the equation. Multiplying the equation by

2

yields

2x^2 - 5x + 2 = 0

. Factoring this expression gives

(2x - 1)(x - 2) = 0

, which yields the solutions

x = \frac{1}{2}

and

x = 2

Multiply the entire equation by

2

to clear the fraction.

2x^2 - 5x + 2 = 0

Converting the equation to integer coefficients simplifies factoring or applying the quadratic formula.

Identify the coefficients

a

b

, and

c

to use in the quadratic formula.

a = 2

b = -5

c = 2

These coefficients correspond to the standard quadratic form

ax^2 + bx + c = 0

Calculate the discriminant

b^2 - 4ac

(-5)^2 - 4(2)(2) = 25 - 16 = 9

The discriminant determines the number and type of solutions.

Substitute the coefficients and the discriminant into the quadratic formula:

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

x = \frac{-(-5) \pm \sqrt{9}}{2(2)} = \frac{5 \pm 3}{4}

This formula provides the exact solutions to any quadratic equation.

Evaluate the two cases for the plus/minus sign.

x = \frac{5 + 3}{4} = 2

and

x = \frac{5 - 3}{4} = \frac{1}{2}

Splitting the formula into positive and negative branches yields the two distinct real roots.

Solving quadratic equations using the quadratic formula

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