Solve 2div(x-1)=4*x+8

$$2 \div ( x - 1 ) = 4 \cdot x + 8$$

Solve for x

$$x = \frac{\sqrt{11} - 1}{2} \approx 1.158312395$$

$$x=\frac{-\sqrt{11}-1}{2}\approx -2.158312395$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$2=4x\left(x-1\right)+\left(x-1\right)\times 8$$

Use the distributive property to multiply 4x by x-1.

$$2=4x^{2}-4x+\left(x-1\right)\times 8$$

Use the distributive property to multiply x-1 by 8.

$$2=4x^{2}-4x+8x-8$$

Combine -4x and 8x to get 4x.

$$2=4x^{2}+4x-8$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}+4x-8=2$$

Subtract 2 from both sides.

$$4x^{2}+4x-8-2=0$$

Subtract 2 from -8 to get -10.

$$4x^{2}+4x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 4 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 4\left(-10\right)}}{2\times 4}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 4\left(-10\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-4±\sqrt{16-16\left(-10\right)}}{2\times 4}$$

Multiply -16 times -10.

$$x=\frac{-4±\sqrt{16+160}}{2\times 4}$$

Add 16 to 160.

$$x=\frac{-4±\sqrt{176}}{2\times 4}$$

Take the square root of 176.

$$x=\frac{-4±4\sqrt{11}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-4±4\sqrt{11}}{8}$$

Now solve the equation x=\frac{-4±4\sqrt{11}}{8} when ± is plus. Add -4 to 4\sqrt{11}.

$$x=\frac{4\sqrt{11}-4}{8}$$

Divide -4+4\sqrt{11} by 8.

$$x=\frac{\sqrt{11}-1}{2}$$

Now solve the equation x=\frac{-4±4\sqrt{11}}{8} when ± is minus. Subtract 4\sqrt{11} from -4.

$$x=\frac{-4\sqrt{11}-4}{8}$$

Divide -4-4\sqrt{11} by 8.

$$x=\frac{-\sqrt{11}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{11}-1}{2} x=\frac{-\sqrt{11}-1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$2=4x\left(x-1\right)+\left(x-1\right)\times 8$$

Use the distributive property to multiply 4x by x-1.

$$2=4x^{2}-4x+\left(x-1\right)\times 8$$

Use the distributive property to multiply x-1 by 8.

$$2=4x^{2}-4x+8x-8$$

Combine -4x and 8x to get 4x.

$$2=4x^{2}+4x-8$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{2}+4x-8=2$$

Add 8 to both sides.

$$4x^{2}+4x=2+8$$

Add 2 and 8 to get 10.

$$4x^{2}+4x=10$$

Divide both sides by 4.

$$\frac{4x^{2}+4x}{4}=\frac{10}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{4}{4}x=\frac{10}{4}$$

Divide 4 by 4.

$$x^{2}+x=\frac{10}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+x=\frac{5}{2}$$

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{5}{2}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{5}{2}+\frac{1}{4}$$

Add \frac{5}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{11}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{11}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{11}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{11}}{2} x+\frac{1}{2}=-\frac{\sqrt{11}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{11}-1}{2} x=\frac{-\sqrt{11}-1}{2}$$