Solve 4x/2x-21/2x-2=1

$$\frac { 4 x } { 2 x } - \frac { 21 } { 2 x - 2 } = 1$$

Solve for x

$$x = \frac{23}{2} = 11\frac{1}{2} = 11.5$$

Steps Using Factoring

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-1\right), the least common multiple of 2x,2x-2.

$$\left(x-1\right)\times 4x-x\times 21=2x\left(x-1\right)$$

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

$$\left(4x-4\right)x-x\times 21=2x\left(x-1\right)$$

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

$$4x^{2}-4x-x\times 21=2x\left(x-1\right)$$

Use the distributive property to multiply 2x by x-1.

$$4x^{2}-4x-x\times 21=2x^{2}-2x$$

Subtract 2x^{2} from both sides.

$$4x^{2}-4x-x\times 21-2x^{2}=-2x$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}-4x-x\times 21=-2x$$

Add 2x to both sides.

$$2x^{2}-4x-x\times 21+2x=0$$

Combine -4x and 2x to get -2x.

$$2x^{2}-2x-x\times 21=0$$

Multiply -1 and 21 to get -21.

$$2x^{2}-2x-21x=0$$

Combine -2x and -21x to get -23x.

$$2x^{2}-23x=0$$

Factor out x.

$$x\left(2x-23\right)=0$$

To find equation solutions, solve x=0 and 2x-23=0.

$$x=0 x=\frac{23}{2}$$

Variable x cannot be equal to 0.

$$x=\frac{23}{2}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-1\right), the least common multiple of 2x,2x-2.

$$\left(x-1\right)\times 4x-x\times 21=2x\left(x-1\right)$$

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

$$\left(4x-4\right)x-x\times 21=2x\left(x-1\right)$$

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

$$4x^{2}-4x-x\times 21=2x\left(x-1\right)$$

Use the distributive property to multiply 2x by x-1.

$$4x^{2}-4x-x\times 21=2x^{2}-2x$$

Subtract 2x^{2} from both sides.

$$4x^{2}-4x-x\times 21-2x^{2}=-2x$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}-4x-x\times 21=-2x$$

Add 2x to both sides.

$$2x^{2}-4x-x\times 21+2x=0$$

Combine -4x and 2x to get -2x.

$$2x^{2}-2x-x\times 21=0$$

Multiply -1 and 21 to get -21.

$$2x^{2}-2x-21x=0$$

Combine -2x and -21x to get -23x.

$$2x^{2}-23x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -23 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}}}{2\times 2}$$

Take the square root of \left(-23\right)^{2}.

$$x=\frac{-\left(-23\right)±23}{2\times 2}$$

The opposite of -23 is 23.

$$x=\frac{23±23}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{23±23}{4}$$

Now solve the equation x=\frac{23±23}{4} when ± is plus. Add 23 to 23.

$$x=\frac{46}{4}$$

Reduce the fraction \frac{46}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{23}{2}$$

Now solve the equation x=\frac{23±23}{4} when ± is minus. Subtract 23 from 23.

$$x=\frac{0}{4}$$

Divide 0 by 4.

$$x=0$$

The equation is now solved.

$$x=\frac{23}{2} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{23}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x-1\right), the least common multiple of 2x,2x-2.

$$\left(x-1\right)\times 4x-x\times 21=2x\left(x-1\right)$$

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

$$\left(4x-4\right)x-x\times 21=2x\left(x-1\right)$$

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

$$4x^{2}-4x-x\times 21=2x\left(x-1\right)$$

Use the distributive property to multiply 2x by x-1.

$$4x^{2}-4x-x\times 21=2x^{2}-2x$$

Subtract 2x^{2} from both sides.

$$4x^{2}-4x-x\times 21-2x^{2}=-2x$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}-4x-x\times 21=-2x$$

Add 2x to both sides.

$$2x^{2}-4x-x\times 21+2x=0$$

Combine -4x and 2x to get -2x.

$$2x^{2}-2x-x\times 21=0$$

Multiply -1 and 21 to get -21.

$$2x^{2}-2x-21x=0$$

Combine -2x and -21x to get -23x.

$$2x^{2}-23x=0$$

Divide both sides by 2.

$$\frac{2x^{2}-23x}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{23}{2}x=\frac{0}{2}$$

Divide 0 by 2.

$$x^{2}-\frac{23}{2}x=0$$

Divide -\frac{23}{2}, the coefficient of the x term, by 2 to get -\frac{23}{4}. Then add the square of -\frac{23}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{2}x+\left(-\frac{23}{4}\right)^{2}=\left(-\frac{23}{4}\right)^{2}$$

Square -\frac{23}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{2}x+\frac{529}{16}=\frac{529}{16}$$

Factor x^{2}-\frac{23}{2}x+\frac{529}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{23}{4}\right)^{2}=\frac{529}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{4}\right)^{2}}=\sqrt{\frac{529}{16}}$$

Simplify.

$$x-\frac{23}{4}=\frac{23}{4} x-\frac{23}{4}=-\frac{23}{4}$$

Add \frac{23}{4} to both sides of the equation.

$$x=\frac{23}{2} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{23}{2}$$