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

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

Solve for x

$$x = \frac{\sqrt{4281} + 85}{92} \approx 1.635101644$$

$$x=\frac{85-\sqrt{4281}}{92}\approx 0.212724443$$

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.

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

Use the distributive property to multiply 4x-7 by 9x+7 and combine like terms.

$$36x^{2}-35x-49=\left(7x-9\right)\left(9-8x\right)$$

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

$$36x^{2}-35x-49=135x-56x^{2}-81$$

Subtract 135x from both sides.

$$36x^{2}-35x-49-135x=-56x^{2}-81$$

Combine -35x and -135x to get -170x.

$$36x^{2}-170x-49=-56x^{2}-81$$

Add 56x^{2} to both sides.

$$36x^{2}-170x-49+56x^{2}=-81$$

Combine 36x^{2} and 56x^{2} to get 92x^{2}.

$$92x^{2}-170x-49=-81$$

Add 81 to both sides.

$$92x^{2}-170x-49+81=0$$

Add -49 and 81 to get 32.

$$92x^{2}-170x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 92 for a, -170 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-170\right)±\sqrt{\left(-170\right)^{2}-4\times 92\times 32}}{2\times 92}$$

Square -170.

$$x=\frac{-\left(-170\right)±\sqrt{28900-4\times 92\times 32}}{2\times 92}$$

Multiply -4 times 92.

$$x=\frac{-\left(-170\right)±\sqrt{28900-368\times 32}}{2\times 92}$$

Multiply -368 times 32.

$$x=\frac{-\left(-170\right)±\sqrt{28900-11776}}{2\times 92}$$

Add 28900 to -11776.

$$x=\frac{-\left(-170\right)±\sqrt{17124}}{2\times 92}$$

Take the square root of 17124.

$$x=\frac{-\left(-170\right)±2\sqrt{4281}}{2\times 92}$$

The opposite of -170 is 170.

$$x=\frac{170±2\sqrt{4281}}{2\times 92}$$

Multiply 2 times 92.

$$x=\frac{170±2\sqrt{4281}}{184}$$

Now solve the equation x=\frac{170±2\sqrt{4281}}{184} when ± is plus. Add 170 to 2\sqrt{4281}.

$$x=\frac{2\sqrt{4281}+170}{184}$$

Divide 170+2\sqrt{4281} by 184.

$$x=\frac{\sqrt{4281}+85}{92}$$

Now solve the equation x=\frac{170±2\sqrt{4281}}{184} when ± is minus. Subtract 2\sqrt{4281} from 170.

$$x=\frac{170-2\sqrt{4281}}{184}$$

Divide 170-2\sqrt{4281} by 184.

$$x=\frac{85-\sqrt{4281}}{92}$$

The equation is now solved.

$$x=\frac{\sqrt{4281}+85}{92} x=\frac{85-\sqrt{4281}}{92}$$

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.

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

Use the distributive property to multiply 4x-7 by 9x+7 and combine like terms.

$$36x^{2}-35x-49=\left(7x-9\right)\left(9-8x\right)$$

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

$$36x^{2}-35x-49=135x-56x^{2}-81$$

Subtract 135x from both sides.

$$36x^{2}-35x-49-135x=-56x^{2}-81$$

Combine -35x and -135x to get -170x.

$$36x^{2}-170x-49=-56x^{2}-81$$

Add 56x^{2} to both sides.

$$36x^{2}-170x-49+56x^{2}=-81$$

Combine 36x^{2} and 56x^{2} to get 92x^{2}.

$$92x^{2}-170x-49=-81$$

Add 49 to both sides.

$$92x^{2}-170x=-81+49$$

Add -81 and 49 to get -32.

$$92x^{2}-170x=-32$$

Divide both sides by 92.

$$\frac{92x^{2}-170x}{92}=-\frac{32}{92}$$

Dividing by 92 undoes the multiplication by 92.

$$x^{2}+\left(-\frac{170}{92}\right)x=-\frac{32}{92}$$

Reduce the fraction \frac{-170}{92} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{85}{46}x=-\frac{32}{92}$$

Reduce the fraction \frac{-32}{92} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{85}{46}x=-\frac{8}{23}$$

Divide -\frac{85}{46}, the coefficient of the x term, by 2 to get -\frac{85}{92}. Then add the square of -\frac{85}{92} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{85}{46}x+\left(-\frac{85}{92}\right)^{2}=-\frac{8}{23}+\left(-\frac{85}{92}\right)^{2}$$

Square -\frac{85}{92} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{85}{46}x+\frac{7225}{8464}=-\frac{8}{23}+\frac{7225}{8464}$$

Add -\frac{8}{23} to \frac{7225}{8464} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{85}{46}x+\frac{7225}{8464}=\frac{4281}{8464}$$

Factor x^{2}-\frac{85}{46}x+\frac{7225}{8464}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{85}{92}\right)^{2}=\frac{4281}{8464}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{85}{92}\right)^{2}}=\sqrt{\frac{4281}{8464}}$$

Simplify.

$$x-\frac{85}{92}=\frac{\sqrt{4281}}{92} x-\frac{85}{92}=-\frac{\sqrt{4281}}{92}$$

Add \frac{85}{92} to both sides of the equation.

$$x=\frac{\sqrt{4281}+85}{92} x=\frac{85-\sqrt{4281}}{92}$$