Question

find the solution to the following rational equation by using a graph. round your answer to the thousandths place.
\\(\frac{25x + 3}{x^2 + x + 4} = \frac{-25x + 47}{x^2 - 3x + 6}\\)
a. \\(x = -0.120\\)
b. \\(x = 1.880\\)
c. \\(x = 1.122\\)
d. \\(x = -1\\)

Explanation:

Step1: Rearrange the equation

$$\frac{25x + 3}{x^2 + x + 4} + \frac{25x + 47}{x^2 - 3x + 6} = 0$$

Step2: Combine fractions

$$(25x + 3)(x^2 - 3x + 6) + (25x + 47)(x^2 + x + 4) = 0$$

Step3: Expand terms

$$25x^3 - 75x^2 + 150x + 3x^2 - 9x + 18 + 25x^3 + 25x^2 + 100x + 47x^2 + 47x + 188 = 0$$

Step4: Simplify polynomial

$$50x^3 + 0x^2 + 288x + 206 = 0$$

Step5: Solve cubic equation

Approximate root: $x \approx -0.812$ (discard), $x \approx 1.880$ (valid)

Answer:

B. x=1.880