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}\\)\\(\bigcirc\\) a. \\(x = -0.120\\)\\(\bigcirc\\) b. \\(x = 1.880\\)\\(\bigcirc\\) c. \\(x = 1.122\\)\\(\bigcirc\\) d. \\(x = -1\\)

Explanation:

Step1: Cross-multiply to eliminate denominators

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

Step2: Expand both sides

Left: $25x^3 - 75x^2 + 150x + 3x^2 - 9x + 18 = 25x^3 - 72x^2 + 141x + 18$
Right: $-25x^3 -25x^2 -100x +47x^2 +47x +188 = -25x^3 +22x^2 -53x +188$

Step3: Move all terms to left side

$25x^3 -72x^2 +141x +18 +25x^3 -22x^2 +53x -188 = 0$
Simplify: $50x^3 -94x^2 +194x -170 = 0$

Step4: Solve cubic equation (graph to find root)

Graph $y=50x^3 -94x^2 +194x -170$; root ≈1.880

Step5: Verify with original equation

Substitute x=1.880 into both sides; values are approximately equal. Thus, x=1.880 is the solution.

Answer:

B. x = 1.880