If the equation 3x^2 + 5x = 2 is solved for x, what is the positive difference between the… — ACT

If the equation $3x^2 + 5x = 2$ is solved for $x$ , what is the positive difference between the two solutions?

The positive difference between the two solutions is

\frac{7}{3}

Rearranging the equation

3x^2 + 5x = 2

by subtracting

2

from both sides gives the standard quadratic equation

3x^2 + 5x - 2 = 0

. Factoring this expression yields

(3x - 1)(x + 2) = 0

. Setting each factor to zero gives the solutions

x = \frac{1}{3}

and

x = -2

. The positive difference between these solutions is

\frac{1}{3} - (-2) = \frac{1}{3} + 2 = \frac{7}{3}

Rearrange the quadratic equation into standard form

ax^2 + bx + c = 0

3x^2 + 5x - 2 = 0

Before factoring a quadratic equation, all terms must be moved to one side so that the equation is set to zero.

Factor the quadratic expression by finding two binomials that multiply to

3x^2 + 5x - 2

(3x - 1)(x + 2) = 0

Since the product of

3

and

-2

-6

, we look for factors of

-6

that sum to

5

, which are

6

and

-1

. Splitting the middle term yields

3x^2 + 6x - x - 2 = 3x(x + 2) - 1(x + 2) = (3x - 1)(x + 2)

Use the zero-product property to solve for

x

by setting each binomial factor to zero.

3x - 1 = 0 \Rightarrow x = \frac{1}{3}

and

x + 2 = 0 \Rightarrow x = -2

If the product of two quantities is zero, at least one of the quantities must be zero.

Calculate the positive difference between the two solutions.

|\frac{1}{3} - (-2)| = |\frac{1}{3} + 2| = \frac{7}{3}

The positive difference is the absolute value of the subtraction of one root from the other.

Solving a quadratic equation by factoring over the integers after setting the equation equal to zero.

Alternatively, the quadratic formula can be used. Once the equation is rewritten as

3x^2 + 5x - 2 = 0

, substitute

a=3

b=5

, and

c=-2

into

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

to get

x = \frac{-5 \pm \sqrt{25 - 4(3)(-2)}}{6} = \frac{-5 \pm 7}{6}

. This simplifies to

x = \frac{1}{3}

and

x = -2

. The positive difference between these values is

\frac{1}{3} - (-2) = \frac{7}{3}

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