Solve x-7/x-1+2x-6/2x-1=5/2

$$\frac{ x-7 }{ x-1 } + \frac{ 2x-6 }{ 2x-1 } = \frac{ 5 }{ 2 }$$

Solve for x

$$x=\frac{\sqrt{1129}-31}{4}\approx 0.650148808$$

$$x=\frac{-\sqrt{1129}-31}{4}\approx -16.150148808$$

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 2\left(x-1\right)\left(2x-1\right), the least common multiple of x-1,2x-1,2.

$$\left(4x-2\right)\left(x-7\right)+\left(2x-2\right)\left(2x-6\right)=5\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 4x-2 by x-7 and combine like terms.

$$4x^{2}-30x+14+\left(2x-2\right)\left(2x-6\right)=5\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-30x+14+4x^{2}-16x+12=5\left(x-1\right)\left(2x-1\right)$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}-30x+14-16x+12=5\left(x-1\right)\left(2x-1\right)$$

Combine -30x and -16x to get -46x.

$$8x^{2}-46x+14+12=5\left(x-1\right)\left(2x-1\right)$$

Add 14 and 12 to get 26.

$$8x^{2}-46x+26=5\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 5 by x-1.

$$8x^{2}-46x+26=\left(5x-5\right)\left(2x-1\right)$$

Use the distributive property to multiply 5x-5 by 2x-1 and combine like terms.

$$8x^{2}-46x+26=10x^{2}-15x+5$$

Subtract 10x^{2} from both sides.

$$8x^{2}-46x+26-10x^{2}=-15x+5$$

Combine 8x^{2} and -10x^{2} to get -2x^{2}.

$$-2x^{2}-46x+26=-15x+5$$

Add 15x to both sides.

$$-2x^{2}-46x+26+15x=5$$

Combine -46x and 15x to get -31x.

$$-2x^{2}-31x+26=5$$

Subtract 5 from both sides.

$$-2x^{2}-31x+26-5=0$$

Subtract 5 from 26 to get 21.

$$-2x^{2}-31x+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -31 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-31\right)±\sqrt{\left(-31\right)^{2}-4\left(-2\right)\times 21}}{2\left(-2\right)}$$

Square -31.

$$x=\frac{-\left(-31\right)±\sqrt{961-4\left(-2\right)\times 21}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-31\right)±\sqrt{961+8\times 21}}{2\left(-2\right)}$$

Multiply 8 times 21.

$$x=\frac{-\left(-31\right)±\sqrt{961+168}}{2\left(-2\right)}$$

Add 961 to 168.

$$x=\frac{-\left(-31\right)±\sqrt{1129}}{2\left(-2\right)}$$

The opposite of -31 is 31.

$$x=\frac{31±\sqrt{1129}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{31±\sqrt{1129}}{-4}$$

Now solve the equation x=\frac{31±\sqrt{1129}}{-4} when ± is plus. Add 31 to \sqrt{1129}.

$$x=\frac{\sqrt{1129}+31}{-4}$$

Divide 31+\sqrt{1129} by -4.

$$x=\frac{-\sqrt{1129}-31}{4}$$

Now solve the equation x=\frac{31±\sqrt{1129}}{-4} when ± is minus. Subtract \sqrt{1129} from 31.

$$x=\frac{31-\sqrt{1129}}{-4}$$

Divide 31-\sqrt{1129} by -4.

$$x=\frac{\sqrt{1129}-31}{4}$$

The equation is now solved.

$$x=\frac{-\sqrt{1129}-31}{4} x=\frac{\sqrt{1129}-31}{4}$$

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 2\left(x-1\right)\left(2x-1\right), the least common multiple of x-1,2x-1,2.

$$\left(4x-2\right)\left(x-7\right)+\left(2x-2\right)\left(2x-6\right)=5\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 4x-2 by x-7 and combine like terms.

$$4x^{2}-30x+14+\left(2x-2\right)\left(2x-6\right)=5\left(x-1\right)\left(2x-1\right)$$

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

$$4x^{2}-30x+14+4x^{2}-16x+12=5\left(x-1\right)\left(2x-1\right)$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}-30x+14-16x+12=5\left(x-1\right)\left(2x-1\right)$$

Combine -30x and -16x to get -46x.

$$8x^{2}-46x+14+12=5\left(x-1\right)\left(2x-1\right)$$

Add 14 and 12 to get 26.

$$8x^{2}-46x+26=5\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 5 by x-1.

$$8x^{2}-46x+26=\left(5x-5\right)\left(2x-1\right)$$

Use the distributive property to multiply 5x-5 by 2x-1 and combine like terms.

$$8x^{2}-46x+26=10x^{2}-15x+5$$

Subtract 10x^{2} from both sides.

$$8x^{2}-46x+26-10x^{2}=-15x+5$$

Combine 8x^{2} and -10x^{2} to get -2x^{2}.

$$-2x^{2}-46x+26=-15x+5$$

Add 15x to both sides.

$$-2x^{2}-46x+26+15x=5$$

Combine -46x and 15x to get -31x.

$$-2x^{2}-31x+26=5$$

Subtract 26 from both sides.

$$-2x^{2}-31x=5-26$$

Subtract 26 from 5 to get -21.

$$-2x^{2}-31x=-21$$

Divide both sides by -2.

$$\frac{-2x^{2}-31x}{-2}=-\frac{21}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{31}{-2}\right)x=-\frac{21}{-2}$$

Divide -31 by -2.

$$x^{2}+\frac{31}{2}x=-\frac{21}{-2}$$

Divide -21 by -2.

$$x^{2}+\frac{31}{2}x=\frac{21}{2}$$

Divide \frac{31}{2}, the coefficient of the x term, by 2 to get \frac{31}{4}. Then add the square of \frac{31}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{31}{2}x+\left(\frac{31}{4}\right)^{2}=\frac{21}{2}+\left(\frac{31}{4}\right)^{2}$$

Square \frac{31}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{31}{2}x+\frac{961}{16}=\frac{21}{2}+\frac{961}{16}$$

Add \frac{21}{2} to \frac{961}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{31}{2}x+\frac{961}{16}=\frac{1129}{16}$$

Factor x^{2}+\frac{31}{2}x+\frac{961}{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}.

$$\left(x+\frac{31}{4}\right)^{2}=\frac{1129}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{31}{4}\right)^{2}}=\sqrt{\frac{1129}{16}}$$

Simplify.

$$x+\frac{31}{4}=\frac{\sqrt{1129}}{4} x+\frac{31}{4}=-\frac{\sqrt{1129}}{4}$$

Subtract \frac{31}{4} from both sides of the equation.

$$x=\frac{\sqrt{1129}-31}{4} x=\frac{-\sqrt{1129}-31}{4}$$