Solve x=-7(x-2)/2x-11

$$x = - \frac { 7 ( x - 2 ) } { 2 x - 11 }$$

Solve for x

$$x=2\sqrt{2}+1\approx 3.828427125$$

$$x=1-2\sqrt{2}\approx -1.828427125$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 7 by x-2.

$$x=-\frac{7x-14}{2x-11}$$

Add \frac{7x-14}{2x-11} to both sides.

$$x+\frac{7x-14}{2x-11}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2x-11}{2x-11}.

$$\frac{x\left(2x-11\right)}{2x-11}+\frac{7x-14}{2x-11}=0$$

Since \frac{x\left(2x-11\right)}{2x-11} and \frac{7x-14}{2x-11} have the same denominator, add them by adding their numerators.

$$\frac{x\left(2x-11\right)+7x-14}{2x-11}=0$$

Do the multiplications in x\left(2x-11\right)+7x-14.

$$\frac{2x^{2}-11x+7x-14}{2x-11}=0$$

Combine like terms in 2x^{2}-11x+7x-14.

$$\frac{2x^{2}-4x-14}{2x-11}=0$$

Variable x cannot be equal to \frac{11}{2} since division by zero is not defined. Multiply both sides of the equation by 2x-11.

$$2x^{2}-4x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-14\right)}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-14\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-14\right)}}{2\times 2}$$

Multiply -8 times -14.

$$x=\frac{-\left(-4\right)±\sqrt{16+112}}{2\times 2}$$

Add 16 to 112.

$$x=\frac{-\left(-4\right)±\sqrt{128}}{2\times 2}$$

Take the square root of 128.

$$x=\frac{-\left(-4\right)±8\sqrt{2}}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±8\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±8\sqrt{2}}{4}$$

Now solve the equation x=\frac{4±8\sqrt{2}}{4} when ± is plus. Add 4 to 8\sqrt{2}.

$$x=\frac{8\sqrt{2}+4}{4}$$

Divide 8\sqrt{2}+4 by 4.

$$x=2\sqrt{2}+1$$

Now solve the equation x=\frac{4±8\sqrt{2}}{4} when ± is minus. Subtract 8\sqrt{2} from 4.

$$x=\frac{4-8\sqrt{2}}{4}$$

Divide 4-8\sqrt{2} by 4.

$$x=1-2\sqrt{2}$$

The equation is now solved.

$$x=2\sqrt{2}+1 x=1-2\sqrt{2}$$

Steps for Completing the Square

Use the distributive property to multiply 7 by x-2.

$$x=-\frac{7x-14}{2x-11}$$

Add \frac{7x-14}{2x-11} to both sides.

$$x+\frac{7x-14}{2x-11}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2x-11}{2x-11}.

$$\frac{x\left(2x-11\right)}{2x-11}+\frac{7x-14}{2x-11}=0$$

Since \frac{x\left(2x-11\right)}{2x-11} and \frac{7x-14}{2x-11} have the same denominator, add them by adding their numerators.

$$\frac{x\left(2x-11\right)+7x-14}{2x-11}=0$$

Do the multiplications in x\left(2x-11\right)+7x-14.

$$\frac{2x^{2}-11x+7x-14}{2x-11}=0$$

Combine like terms in 2x^{2}-11x+7x-14.

$$\frac{2x^{2}-4x-14}{2x-11}=0$$

Variable x cannot be equal to \frac{11}{2} since division by zero is not defined. Multiply both sides of the equation by 2x-11.

$$2x^{2}-4x-14=0$$

Add 14 to both sides. Anything plus zero gives itself.

$$2x^{2}-4x=14$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{14}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{14}{2}$$

Divide -4 by 2.

$$x^{2}-2x=\frac{14}{2}$$

Divide 14 by 2.

$$x^{2}-2x=7$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=7+1$$

Add 7 to 1.

$$x^{2}-2x+1=8$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=8$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{8}$$

Simplify.

$$x-1=2\sqrt{2} x-1=-2\sqrt{2}$$

Add 1 to both sides of the equation.

$$x=2\sqrt{2}+1 x=1-2\sqrt{2}$$