Solve x-7/x-1+2x-6/2x-1=0

$$\frac{ x-7 }{ x-1 } + \frac{ 2x-6 }{ 2x-1 } = 0$$

Solve for x

$$x = \frac{\sqrt{321} + 23}{8} \approx 5.114559108$$

$$x=\frac{23-\sqrt{321}}{8}\approx 0.635440892$$

Steps Using the Quadratic Formula

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

$$\left(2x-1\right)\left(x-7\right)+\left(x-1\right)\left(2x-6\right)=0$$

Use the distributive property to multiply 2x-1 by x-7 and combine like terms.

$$2x^{2}-15x+7+\left(x-1\right)\left(2x-6\right)=0$$

Use the distributive property to multiply x-1 by 2x-6 and combine like terms.

$$2x^{2}-15x+7+2x^{2}-8x+6=0$$

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

$$4x^{2}-15x+7-8x+6=0$$

Combine -15x and -8x to get -23x.

$$4x^{2}-23x+7+6=0$$

Add 7 and 6 to get 13.

$$4x^{2}-23x+13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -23 for b, and 13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 4\times 13}}{2\times 4}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\times 4\times 13}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-23\right)±\sqrt{529-16\times 13}}{2\times 4}$$

Multiply -16 times 13.

$$x=\frac{-\left(-23\right)±\sqrt{529-208}}{2\times 4}$$

Add 529 to -208.

$$x=\frac{-\left(-23\right)±\sqrt{321}}{2\times 4}$$

The opposite of -23 is 23.

$$x=\frac{23±\sqrt{321}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{23±\sqrt{321}}{8}$$

Now solve the equation x=\frac{23±\sqrt{321}}{8} when ± is plus. Add 23 to \sqrt{321}.

$$x=\frac{\sqrt{321}+23}{8}$$

Now solve the equation x=\frac{23±\sqrt{321}}{8} when ± is minus. Subtract \sqrt{321} from 23.

$$x=\frac{23-\sqrt{321}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{321}+23}{8} x=\frac{23-\sqrt{321}}{8}$$

Steps for Completing the Square

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

$$\left(2x-1\right)\left(x-7\right)+\left(x-1\right)\left(2x-6\right)=0$$

Use the distributive property to multiply 2x-1 by x-7 and combine like terms.

$$2x^{2}-15x+7+\left(x-1\right)\left(2x-6\right)=0$$

Use the distributive property to multiply x-1 by 2x-6 and combine like terms.

$$2x^{2}-15x+7+2x^{2}-8x+6=0$$

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

$$4x^{2}-15x+7-8x+6=0$$

Combine -15x and -8x to get -23x.

$$4x^{2}-23x+7+6=0$$

Add 7 and 6 to get 13.

$$4x^{2}-23x+13=0$$

Subtract 13 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-23x=-13$$

Divide both sides by 4.

$$\frac{4x^{2}-23x}{4}=-\frac{13}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{23}{4}x=-\frac{13}{4}$$

Divide -\frac{23}{4}, the coefficient of the x term, by 2 to get -\frac{23}{8}. Then add the square of -\frac{23}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{4}x+\left(-\frac{23}{8}\right)^{2}=-\frac{13}{4}+\left(-\frac{23}{8}\right)^{2}$$

Square -\frac{23}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{4}x+\frac{529}{64}=-\frac{13}{4}+\frac{529}{64}$$

Add -\frac{13}{4} to \frac{529}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{23}{4}x+\frac{529}{64}=\frac{321}{64}$$

Factor x^{2}-\frac{23}{4}x+\frac{529}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{23}{8}\right)^{2}=\frac{321}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{8}\right)^{2}}=\sqrt{\frac{321}{64}}$$

Simplify.

$$x-\frac{23}{8}=\frac{\sqrt{321}}{8} x-\frac{23}{8}=-\frac{\sqrt{321}}{8}$$

Add \frac{23}{8} to both sides of the equation.

$$x=\frac{\sqrt{321}+23}{8} x=\frac{23-\sqrt{321}}{8}$$