Solve 8x+7/7x-9=9-8x/4x-7

$$\frac { 8 x + 7 } { 7 x - 9 } = \frac { 9 - 8 x } { 4 x - 7 }$$

Solve for x

$$x = \frac{\sqrt{15305} + 163}{176} \approx 1.629053286$$

$$x=\frac{163-\sqrt{15305}}{176}\approx 0.223219441$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{9}{7},\frac{7}{4} since division by zero is not defined. Multiply both sides of the equation by \left(4x-7\right)\left(7x-9\right), the least common multiple of 7x-9,4x-7.

$$\left(4x-7\right)\left(8x+7\right)=\left(7x-9\right)\left(9-8x\right)$$

Use the distributive property to multiply 4x-7 by 8x+7 and combine like terms.

$$32x^{2}-28x-49=\left(7x-9\right)\left(9-8x\right)$$

Use the distributive property to multiply 7x-9 by 9-8x and combine like terms.

$$32x^{2}-28x-49=135x-56x^{2}-81$$

Subtract 135x from both sides.

$$32x^{2}-28x-49-135x=-56x^{2}-81$$

Combine -28x and -135x to get -163x.

$$32x^{2}-163x-49=-56x^{2}-81$$

Add 56x^{2} to both sides.

$$32x^{2}-163x-49+56x^{2}=-81$$

Combine 32x^{2} and 56x^{2} to get 88x^{2}.

$$88x^{2}-163x-49=-81$$

Add 81 to both sides.

$$88x^{2}-163x-49+81=0$$

Add -49 and 81 to get 32.

$$88x^{2}-163x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 88 for a, -163 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-163\right)±\sqrt{\left(-163\right)^{2}-4\times 88\times 32}}{2\times 88}$$

Square -163.

$$x=\frac{-\left(-163\right)±\sqrt{26569-4\times 88\times 32}}{2\times 88}$$

Multiply -4 times 88.

$$x=\frac{-\left(-163\right)±\sqrt{26569-352\times 32}}{2\times 88}$$

Multiply -352 times 32.

$$x=\frac{-\left(-163\right)±\sqrt{26569-11264}}{2\times 88}$$

Add 26569 to -11264.

$$x=\frac{-\left(-163\right)±\sqrt{15305}}{2\times 88}$$

The opposite of -163 is 163.

$$x=\frac{163±\sqrt{15305}}{2\times 88}$$

Multiply 2 times 88.

$$x=\frac{163±\sqrt{15305}}{176}$$

Now solve the equation x=\frac{163±\sqrt{15305}}{176} when ± is plus. Add 163 to \sqrt{15305}.

$$x=\frac{\sqrt{15305}+163}{176}$$

Now solve the equation x=\frac{163±\sqrt{15305}}{176} when ± is minus. Subtract \sqrt{15305} from 163.

$$x=\frac{163-\sqrt{15305}}{176}$$

The equation is now solved.

$$x=\frac{\sqrt{15305}+163}{176} x=\frac{163-\sqrt{15305}}{176}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{9}{7},\frac{7}{4} since division by zero is not defined. Multiply both sides of the equation by \left(4x-7\right)\left(7x-9\right), the least common multiple of 7x-9,4x-7.

$$\left(4x-7\right)\left(8x+7\right)=\left(7x-9\right)\left(9-8x\right)$$

Use the distributive property to multiply 4x-7 by 8x+7 and combine like terms.

$$32x^{2}-28x-49=\left(7x-9\right)\left(9-8x\right)$$

Use the distributive property to multiply 7x-9 by 9-8x and combine like terms.

$$32x^{2}-28x-49=135x-56x^{2}-81$$

Subtract 135x from both sides.

$$32x^{2}-28x-49-135x=-56x^{2}-81$$

Combine -28x and -135x to get -163x.

$$32x^{2}-163x-49=-56x^{2}-81$$

Add 56x^{2} to both sides.

$$32x^{2}-163x-49+56x^{2}=-81$$

Combine 32x^{2} and 56x^{2} to get 88x^{2}.

$$88x^{2}-163x-49=-81$$

Add 49 to both sides.

$$88x^{2}-163x=-81+49$$

Add -81 and 49 to get -32.

$$88x^{2}-163x=-32$$

Divide both sides by 88.

$$\frac{88x^{2}-163x}{88}=-\frac{32}{88}$$

Dividing by 88 undoes the multiplication by 88.

$$x^{2}-\frac{163}{88}x=-\frac{32}{88}$$

Reduce the fraction \frac{-32}{88} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{163}{88}x=-\frac{4}{11}$$

Divide -\frac{163}{88}, the coefficient of the x term, by 2 to get -\frac{163}{176}. Then add the square of -\frac{163}{176} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{163}{88}x+\left(-\frac{163}{176}\right)^{2}=-\frac{4}{11}+\left(-\frac{163}{176}\right)^{2}$$

Square -\frac{163}{176} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{163}{88}x+\frac{26569}{30976}=-\frac{4}{11}+\frac{26569}{30976}$$

Add -\frac{4}{11} to \frac{26569}{30976} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{163}{88}x+\frac{26569}{30976}=\frac{15305}{30976}$$

Factor x^{2}-\frac{163}{88}x+\frac{26569}{30976}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{163}{176}\right)^{2}=\frac{15305}{30976}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{163}{176}\right)^{2}}=\sqrt{\frac{15305}{30976}}$$

Simplify.

$$x-\frac{163}{176}=\frac{\sqrt{15305}}{176} x-\frac{163}{176}=-\frac{\sqrt{15305}}{176}$$

Add \frac{163}{176} to both sides of the equation.

$$x=\frac{\sqrt{15305}+163}{176} x=\frac{163-\sqrt{15305}}{176}$$