Question

guided practice
find the solutions to the following rational equation by using a graph. round your answers to the thousandths place.
\\(\frac{x^2 + 3x}{x^3 - 2x + 4} = \frac{19x - 30}{x^3 - 4x + 6}\\)
a. \\(x = -1.951\\) and \\(x = 2.174\\)
b. \\(x = 3.000\\) and \\(x = 13.000\\)
c. \\(x = -2.079\\), \\(x = -1.307\\), and \\(x = 1.495\\)
d. \\(x = -1.000\\) and \\(x = 1.000\\)

Explanation:

Step1: Cross-multiply to eliminate denominators

\((x^2 + 3x)(x^2 - 4x + 6) = (10x - 39)(x^3 - 2x^2 + 4)\)

Step2: Expand both sides

Left: \(x^4 - 4x^3 + 6x^2 + 3x^3 - 12x^2 + 18x = x^4 - x^3 - 6x^2 + 18x\)
Right: \(10x^4 - 20x^3 + 40x - 39x^3 + 78x^2 - 156 = 10x^4 - 59x^3 + 78x^2 + 40x - 156\)

Step3: Rearrange to standard polynomial form

\(9x^4 - 58x^3 + 84x^2 + 22x - 156 = 0\)

Step4: Solve polynomial (graphically)

Roots approx: \(x \approx -1.951\) and \(x \approx 2.174\) (matches option A)

Answer:

A. \(x = -1.951\) and \(x = 2.174\)