Solve x-7/x-1+2x-6/2x-1=5/3

$$\frac{ x-7 }{ x-1 } + \frac{ 2x-6 }{ 2x-1 } = \frac{ 5 }{ 3 }$$

Solve for x

$$x = \frac{\sqrt{661} + 27}{2} \approx 26.354960132$$

$$x=\frac{27-\sqrt{661}}{2}\approx 0.645039868$$

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 3\left(x-1\right)\left(2x-1\right), the least common multiple of x-1,2x-1,3.

$$\left(6x-3\right)\left(x-7\right)+\left(3x-3\right)\left(2x-6\right)=5\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 6x-3 by x-7 and combine like terms.

$$6x^{2}-45x+21+\left(3x-3\right)\left(2x-6\right)=5\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 3x-3 by 2x-6 and combine like terms.

$$6x^{2}-45x+21+6x^{2}-24x+18=5\left(x-1\right)\left(2x-1\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}-45x+21-24x+18=5\left(x-1\right)\left(2x-1\right)$$

Combine -45x and -24x to get -69x.

$$12x^{2}-69x+21+18=5\left(x-1\right)\left(2x-1\right)$$

Add 21 and 18 to get 39.

$$12x^{2}-69x+39=5\left(x-1\right)\left(2x-1\right)$$

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

$$12x^{2}-69x+39=\left(5x-5\right)\left(2x-1\right)$$

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

$$12x^{2}-69x+39=10x^{2}-15x+5$$

Subtract 10x^{2} from both sides.

$$12x^{2}-69x+39-10x^{2}=-15x+5$$

Combine 12x^{2} and -10x^{2} to get 2x^{2}.

$$2x^{2}-69x+39=-15x+5$$

Add 15x to both sides.

$$2x^{2}-69x+39+15x=5$$

Combine -69x and 15x to get -54x.

$$2x^{2}-54x+39=5$$

Subtract 5 from both sides.

$$2x^{2}-54x+39-5=0$$

Subtract 5 from 39 to get 34.

$$2x^{2}-54x+34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -54 for b, and 34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 2\times 34}}{2\times 2}$$

Square -54.

$$x=\frac{-\left(-54\right)±\sqrt{2916-4\times 2\times 34}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-54\right)±\sqrt{2916-8\times 34}}{2\times 2}$$

Multiply -8 times 34.

$$x=\frac{-\left(-54\right)±\sqrt{2916-272}}{2\times 2}$$

Add 2916 to -272.

$$x=\frac{-\left(-54\right)±\sqrt{2644}}{2\times 2}$$

Take the square root of 2644.

$$x=\frac{-\left(-54\right)±2\sqrt{661}}{2\times 2}$$

The opposite of -54 is 54.

$$x=\frac{54±2\sqrt{661}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{54±2\sqrt{661}}{4}$$

Now solve the equation x=\frac{54±2\sqrt{661}}{4} when ± is plus. Add 54 to 2\sqrt{661}.

$$x=\frac{2\sqrt{661}+54}{4}$$

Divide 54+2\sqrt{661} by 4.

$$x=\frac{\sqrt{661}+27}{2}$$

Now solve the equation x=\frac{54±2\sqrt{661}}{4} when ± is minus. Subtract 2\sqrt{661} from 54.

$$x=\frac{54-2\sqrt{661}}{4}$$

Divide 54-2\sqrt{661} by 4.

$$x=\frac{27-\sqrt{661}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{661}+27}{2} x=\frac{27-\sqrt{661}}{2}$$

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 3\left(x-1\right)\left(2x-1\right), the least common multiple of x-1,2x-1,3.

$$\left(6x-3\right)\left(x-7\right)+\left(3x-3\right)\left(2x-6\right)=5\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 6x-3 by x-7 and combine like terms.

$$6x^{2}-45x+21+\left(3x-3\right)\left(2x-6\right)=5\left(x-1\right)\left(2x-1\right)$$

Use the distributive property to multiply 3x-3 by 2x-6 and combine like terms.

$$6x^{2}-45x+21+6x^{2}-24x+18=5\left(x-1\right)\left(2x-1\right)$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}-45x+21-24x+18=5\left(x-1\right)\left(2x-1\right)$$

Combine -45x and -24x to get -69x.

$$12x^{2}-69x+21+18=5\left(x-1\right)\left(2x-1\right)$$

Add 21 and 18 to get 39.

$$12x^{2}-69x+39=5\left(x-1\right)\left(2x-1\right)$$

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

$$12x^{2}-69x+39=\left(5x-5\right)\left(2x-1\right)$$

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

$$12x^{2}-69x+39=10x^{2}-15x+5$$

Subtract 10x^{2} from both sides.

$$12x^{2}-69x+39-10x^{2}=-15x+5$$

Combine 12x^{2} and -10x^{2} to get 2x^{2}.

$$2x^{2}-69x+39=-15x+5$$

Add 15x to both sides.

$$2x^{2}-69x+39+15x=5$$

Combine -69x and 15x to get -54x.

$$2x^{2}-54x+39=5$$

Subtract 39 from both sides.

$$2x^{2}-54x=5-39$$

Subtract 39 from 5 to get -34.

$$2x^{2}-54x=-34$$

Divide both sides by 2.

$$\frac{2x^{2}-54x}{2}=-\frac{34}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{54}{2}\right)x=-\frac{34}{2}$$

Divide -54 by 2.

$$x^{2}-27x=-\frac{34}{2}$$

Divide -34 by 2.

$$x^{2}-27x=-17$$

Divide -27, the coefficient of the x term, by 2 to get -\frac{27}{2}. Then add the square of -\frac{27}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-27x+\left(-\frac{27}{2}\right)^{2}=-17+\left(-\frac{27}{2}\right)^{2}$$

Square -\frac{27}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-27x+\frac{729}{4}=-17+\frac{729}{4}$$

Add -17 to \frac{729}{4}.

$$x^{2}-27x+\frac{729}{4}=\frac{661}{4}$$

Factor x^{2}-27x+\frac{729}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{27}{2}\right)^{2}=\frac{661}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{2}\right)^{2}}=\sqrt{\frac{661}{4}}$$

Simplify.

$$x-\frac{27}{2}=\frac{\sqrt{661}}{2} x-\frac{27}{2}=-\frac{\sqrt{661}}{2}$$

Add \frac{27}{2} to both sides of the equation.

$$x=\frac{\sqrt{661}+27}{2} x=\frac{27-\sqrt{661}}{2}$$