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

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

Solve for x

$$x=\sqrt{19}+5\approx 9.358898944$$

$$x=5-\sqrt{19}\approx 0.641101056$$

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.

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

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

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

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

$$2x^{2}-15x+7+2x^{2}-8x+6=\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-15x+7-8x+6=\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-23x+7+6=\left(x-1\right)\left(2x-1\right)$$

Add 7 and 6 to get 13.

$$4x^{2}-23x+13=\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-23x+13=2x^{2}-3x+1$$

Subtract 2x^{2} from both sides.

$$4x^{2}-23x+13-2x^{2}=-3x+1$$

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

$$2x^{2}-23x+13=-3x+1$$

Add 3x to both sides.

$$2x^{2}-23x+13+3x=1$$

Combine -23x and 3x to get -20x.

$$2x^{2}-20x+13=1$$

Subtract 1 from both sides.

$$2x^{2}-20x+13-1=0$$

Subtract 1 from 13 to get 12.

$$2x^{2}-20x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -20 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 2\times 12}}{2\times 2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 2\times 12}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-20\right)±\sqrt{400-8\times 12}}{2\times 2}$$

Multiply -8 times 12.

$$x=\frac{-\left(-20\right)±\sqrt{400-96}}{2\times 2}$$

Add 400 to -96.

$$x=\frac{-\left(-20\right)±\sqrt{304}}{2\times 2}$$

Take the square root of 304.

$$x=\frac{-\left(-20\right)±4\sqrt{19}}{2\times 2}$$

The opposite of -20 is 20.

$$x=\frac{20±4\sqrt{19}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{20±4\sqrt{19}}{4}$$

Now solve the equation x=\frac{20±4\sqrt{19}}{4} when ± is plus. Add 20 to 4\sqrt{19}.

$$x=\frac{4\sqrt{19}+20}{4}$$

Divide 20+4\sqrt{19} by 4.

$$x=\sqrt{19}+5$$

Now solve the equation x=\frac{20±4\sqrt{19}}{4} when ± is minus. Subtract 4\sqrt{19} from 20.

$$x=\frac{20-4\sqrt{19}}{4}$$

Divide 20-4\sqrt{19} by 4.

$$x=5-\sqrt{19}$$

The equation is now solved.

$$x=\sqrt{19}+5 x=5-\sqrt{19}$$

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.

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

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

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

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

$$2x^{2}-15x+7+2x^{2}-8x+6=\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-15x+7-8x+6=\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-23x+7+6=\left(x-1\right)\left(2x-1\right)$$

Add 7 and 6 to get 13.

$$4x^{2}-23x+13=\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-23x+13=2x^{2}-3x+1$$

Subtract 2x^{2} from both sides.

$$4x^{2}-23x+13-2x^{2}=-3x+1$$

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

$$2x^{2}-23x+13=-3x+1$$

Add 3x to both sides.

$$2x^{2}-23x+13+3x=1$$

Combine -23x and 3x to get -20x.

$$2x^{2}-20x+13=1$$

Subtract 13 from both sides.

$$2x^{2}-20x=1-13$$

Subtract 13 from 1 to get -12.

$$2x^{2}-20x=-12$$

Divide both sides by 2.

$$\frac{2x^{2}-20x}{2}=-\frac{12}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{20}{2}\right)x=-\frac{12}{2}$$

Divide -20 by 2.

$$x^{2}-10x=-\frac{12}{2}$$

Divide -12 by 2.

$$x^{2}-10x=-6$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-6+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-6+25$$

Add -6 to 25.

$$x^{2}-10x+25=19$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=19$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{19}$$

Simplify.

$$x-5=\sqrt{19} x-5=-\sqrt{19}$$

Add 5 to both sides of the equation.

$$x=\sqrt{19}+5 x=5-\sqrt{19}$$