Solve x-7/x-1+2x-6/2x-1=10

$$\frac{ x-7 }{ x-1 } + \frac{ 2x-6 }{ 2x-1 } = 10$$

Solve for x

$$x=\frac{\sqrt{241}+7}{32}\approx 0.703880459$$

$$x=\frac{7-\sqrt{241}}{32}\approx -0.266380459$$

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.

$$\left(2x-1\right)\left(x-7\right)+\left(x-1\right)\left(2x-6\right)=10\left(x-1\right)\left(2x-1\right)$$

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

$$2x^{2}-15x+7+\left(x-1\right)\left(2x-6\right)=10\left(x-1\right)\left(2x-1\right)$$

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

$$2x^{2}-15x+7+2x^{2}-8x+6=10\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-15x+7-8x+6=10\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-23x+7+6=10\left(x-1\right)\left(2x-1\right)$$

Add 7 and 6 to get 13.

$$4x^{2}-23x+13=10\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 10 by x-1.

$$4x^{2}-23x+13=\left(10x-10\right)\left(2x-1\right)$$

Use the distributive property to multiply 10x-10 by 2x-1 and combine like terms.

$$4x^{2}-23x+13=20x^{2}-30x+10$$

Subtract 20x^{2} from both sides.

$$4x^{2}-23x+13-20x^{2}=-30x+10$$

Combine 4x^{2} and -20x^{2} to get -16x^{2}.

$$-16x^{2}-23x+13=-30x+10$$

Add 30x to both sides.

$$-16x^{2}-23x+13+30x=10$$

Combine -23x and 30x to get 7x.

$$-16x^{2}+7x+13=10$$

Subtract 10 from both sides.

$$-16x^{2}+7x+13-10=0$$

Subtract 10 from 13 to get 3.

$$-16x^{2}+7x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 7 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-16\right)\times 3}}{2\left(-16\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-16\right)\times 3}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-7±\sqrt{49+64\times 3}}{2\left(-16\right)}$$

Multiply 64 times 3.

$$x=\frac{-7±\sqrt{49+192}}{2\left(-16\right)}$$

Add 49 to 192.

$$x=\frac{-7±\sqrt{241}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-7±\sqrt{241}}{-32}$$

Now solve the equation x=\frac{-7±\sqrt{241}}{-32} when ± is plus. Add -7 to \sqrt{241}.

$$x=\frac{\sqrt{241}-7}{-32}$$

Divide -7+\sqrt{241} by -32.

$$x=\frac{7-\sqrt{241}}{32}$$

Now solve the equation x=\frac{-7±\sqrt{241}}{-32} when ± is minus. Subtract \sqrt{241} from -7.

$$x=\frac{-\sqrt{241}-7}{-32}$$

Divide -7-\sqrt{241} by -32.

$$x=\frac{\sqrt{241}+7}{32}$$

The equation is now solved.

$$x=\frac{7-\sqrt{241}}{32} x=\frac{\sqrt{241}+7}{32}$$

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.

$$\left(2x-1\right)\left(x-7\right)+\left(x-1\right)\left(2x-6\right)=10\left(x-1\right)\left(2x-1\right)$$

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

$$2x^{2}-15x+7+\left(x-1\right)\left(2x-6\right)=10\left(x-1\right)\left(2x-1\right)$$

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

$$2x^{2}-15x+7+2x^{2}-8x+6=10\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-15x+7-8x+6=10\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-23x+7+6=10\left(x-1\right)\left(2x-1\right)$$

Add 7 and 6 to get 13.

$$4x^{2}-23x+13=10\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 10 by x-1.

$$4x^{2}-23x+13=\left(10x-10\right)\left(2x-1\right)$$

Use the distributive property to multiply 10x-10 by 2x-1 and combine like terms.

$$4x^{2}-23x+13=20x^{2}-30x+10$$

Subtract 20x^{2} from both sides.

$$4x^{2}-23x+13-20x^{2}=-30x+10$$

Combine 4x^{2} and -20x^{2} to get -16x^{2}.

$$-16x^{2}-23x+13=-30x+10$$

Add 30x to both sides.

$$-16x^{2}-23x+13+30x=10$$

Combine -23x and 30x to get 7x.

$$-16x^{2}+7x+13=10$$

Subtract 13 from both sides.

$$-16x^{2}+7x=10-13$$

Subtract 13 from 10 to get -3.

$$-16x^{2}+7x=-3$$

Divide both sides by -16.

$$\frac{-16x^{2}+7x}{-16}=-\frac{3}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{7}{-16}x=-\frac{3}{-16}$$

Divide 7 by -16.

$$x^{2}-\frac{7}{16}x=-\frac{3}{-16}$$

Divide -3 by -16.

$$x^{2}-\frac{7}{16}x=\frac{3}{16}$$

Divide -\frac{7}{16}, the coefficient of the x term, by 2 to get -\frac{7}{32}. Then add the square of -\frac{7}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{16}x+\left(-\frac{7}{32}\right)^{2}=\frac{3}{16}+\left(-\frac{7}{32}\right)^{2}$$

Square -\frac{7}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{16}x+\frac{49}{1024}=\frac{3}{16}+\frac{49}{1024}$$

Add \frac{3}{16} to \frac{49}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{16}x+\frac{49}{1024}=\frac{241}{1024}$$

Factor x^{2}-\frac{7}{16}x+\frac{49}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{32}\right)^{2}=\frac{241}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{32}\right)^{2}}=\sqrt{\frac{241}{1024}}$$

Simplify.

$$x-\frac{7}{32}=\frac{\sqrt{241}}{32} x-\frac{7}{32}=-\frac{\sqrt{241}}{32}$$

Add \frac{7}{32} to both sides of the equation.

$$x=\frac{\sqrt{241}+7}{32} x=\frac{7-\sqrt{241}}{32}$$