Solve 9x-13/8=-3x+5/-9x

$$\frac { 9 x - 13 } { 8 } = \frac { - 3 x + 5 } { - 9 x }$$

Solve for x

$$x = \frac{\sqrt{769} + 47}{54} \approx 1.383904616$$

$$x=\frac{47-\sqrt{769}}{54}\approx 0.356836125$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 72x, the least common multiple of 8,-9x.

$$9x\left(9x-13\right)=-8\left(-3x+5\right)$$

Use the distributive property to multiply 9x by 9x-13.

$$81x^{2}-117x=-8\left(-3x+5\right)$$

Use the distributive property to multiply -8 by -3x+5.

$$81x^{2}-117x=24x-40$$

Subtract 24x from both sides.

$$81x^{2}-117x-24x=-40$$

Combine -117x and -24x to get -141x.

$$81x^{2}-141x=-40$$

Add 40 to both sides.

$$81x^{2}-141x+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 81 for a, -141 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-141\right)±\sqrt{\left(-141\right)^{2}-4\times 81\times 40}}{2\times 81}$$

Square -141.

$$x=\frac{-\left(-141\right)±\sqrt{19881-4\times 81\times 40}}{2\times 81}$$

Multiply -4 times 81.

$$x=\frac{-\left(-141\right)±\sqrt{19881-324\times 40}}{2\times 81}$$

Multiply -324 times 40.

$$x=\frac{-\left(-141\right)±\sqrt{19881-12960}}{2\times 81}$$

Add 19881 to -12960.

$$x=\frac{-\left(-141\right)±\sqrt{6921}}{2\times 81}$$

Take the square root of 6921.

$$x=\frac{-\left(-141\right)±3\sqrt{769}}{2\times 81}$$

The opposite of -141 is 141.

$$x=\frac{141±3\sqrt{769}}{2\times 81}$$

Multiply 2 times 81.

$$x=\frac{141±3\sqrt{769}}{162}$$

Now solve the equation x=\frac{141±3\sqrt{769}}{162} when ± is plus. Add 141 to 3\sqrt{769}.

$$x=\frac{3\sqrt{769}+141}{162}$$

Divide 141+3\sqrt{769} by 162.

$$x=\frac{\sqrt{769}+47}{54}$$

Now solve the equation x=\frac{141±3\sqrt{769}}{162} when ± is minus. Subtract 3\sqrt{769} from 141.

$$x=\frac{141-3\sqrt{769}}{162}$$

Divide 141-3\sqrt{769} by 162.

$$x=\frac{47-\sqrt{769}}{54}$$

The equation is now solved.

$$x=\frac{\sqrt{769}+47}{54} x=\frac{47-\sqrt{769}}{54}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 72x, the least common multiple of 8,-9x.

$$9x\left(9x-13\right)=-8\left(-3x+5\right)$$

Use the distributive property to multiply 9x by 9x-13.

$$81x^{2}-117x=-8\left(-3x+5\right)$$

Use the distributive property to multiply -8 by -3x+5.

$$81x^{2}-117x=24x-40$$

Subtract 24x from both sides.

$$81x^{2}-117x-24x=-40$$

Combine -117x and -24x to get -141x.

$$81x^{2}-141x=-40$$

Divide both sides by 81.

$$\frac{81x^{2}-141x}{81}=-\frac{40}{81}$$

Dividing by 81 undoes the multiplication by 81.

$$x^{2}+\left(-\frac{141}{81}\right)x=-\frac{40}{81}$$

Reduce the fraction \frac{-141}{81} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{47}{27}x=-\frac{40}{81}$$

Divide -\frac{47}{27}, the coefficient of the x term, by 2 to get -\frac{47}{54}. Then add the square of -\frac{47}{54} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{47}{27}x+\left(-\frac{47}{54}\right)^{2}=-\frac{40}{81}+\left(-\frac{47}{54}\right)^{2}$$

Square -\frac{47}{54} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{47}{27}x+\frac{2209}{2916}=-\frac{40}{81}+\frac{2209}{2916}$$

Add -\frac{40}{81} to \frac{2209}{2916} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{47}{27}x+\frac{2209}{2916}=\frac{769}{2916}$$

Factor x^{2}-\frac{47}{27}x+\frac{2209}{2916}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{47}{54}\right)^{2}=\frac{769}{2916}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{47}{54}\right)^{2}}=\sqrt{\frac{769}{2916}}$$

Simplify.

$$x-\frac{47}{54}=\frac{\sqrt{769}}{54} x-\frac{47}{54}=-\frac{\sqrt{769}}{54}$$

Add \frac{47}{54} to both sides of the equation.

$$x=\frac{\sqrt{769}+47}{54} x=\frac{47-\sqrt{769}}{54}$$