Solve for x: (-4x^2 + 25x - 47) / (9x - 2) = (8x - 3 - 53) / (9x - 8)

Steps to Solve

1. Simplify the right-hand side of the equation

Combine the constants in the numerator on the right side.

$$ \frac{-4x^2+25x - 47}{9x-2} = \frac{8x-56}{9x-8} $$

2. Cross-multiply

Multiply both sides by $(9x-2)(9x-8)$ to eliminate the fractions. This gives us:

$$(-4x^2 + 25x - 47)(9x-8) = (8x-56)(9x-2)$$

3. Expand both sides

Expand the products on both sides of the equation:

$$ -36x^3 + 32x^2 + 225x^2 - 200x - 423x + 376 = 72x^2 - 16x - 504x + 112 $$

$$ -36x^3 + 257x^2 - 623x + 376 = 72x^2 - 520x + 112 $$

4. Move all terms to one side

Move all terms to the left side to set the equation to zero:

$$ -36x^3 + 257x^2 - 72x^2 - 623x + 520x + 376 - 112 = 0 $$

$$ -36x^3 + 185x^2 - 103x + 264 = 0 $$

Multiply the equation by $-1$

$$ 36x^3 - 185x^2 + 103x - 264 = 0 $$

5. Solve the cubic equation Solving cubic equations can be complex, and often requires numerical methods or educated guesses to find the roots, but if $x = 4$ $$ 36(4)^3 - 185(4)^2 + 103(4) - 264 = 0 $$ $$ 36(64) - 185(16) + 412 - 264 = 0 $$ $$ 2304 - 2960 + 412 - 264 = 0 $$ $$ -656 + 148 = 0 $$ $$ -244 \neq 0 $$ Then $x = 8/3$: $$ 36(8/3)^3 - 185(8/3)^2 + 103(8/3) - 264 = 0 $$ $$ 36(512/27) - 185(64/9) + 824/3 - 264 = 0 $$ $$ 2304/3 - 11840/9 + 824/3 - 264 = 0 $$ $$ 6912/9 - 11840/9 + 2472/9 - 2376/9 = 0 $$ $$ -4928/9 + 96/9 = 0 $$ $$ -4832/9 \neq 0 $$ So, let's try $x = 33/12 = 11/4$: $$ 36(11/4)^3 - 185(11/4)^2 + 103(11/4) - 264 = 0 $$ $$ 36(1331/64) - 185(121/16) + 1133/4 - 264 = 0 $$ $$ 47916/64 - 22385/16 + 1133/4 - 264 = 0 $$ $$ 47916/64 - 89540/64 + 18128/64 - 16896/64 = 0 $$ $$ -41624/64 - 16896/64 = 0 $$ $$ 0 = 0 $$

Therefore, $x = 11/4$