Solve (x-2)/(x-3)-(x-4)/(x-5)=10/3

$$\frac { ( x - 2 ) } { ( x - 3 ) } - \frac { ( x - 4 ) } { ( x - 5 ) } = \frac { 10 } { 3 }$$

Solve for x

$$x=\frac{\sqrt{10}}{5}+4\approx 4.632455532$$

$$x=-\frac{\sqrt{10}}{5}+4\approx 3.367544468$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 3,5 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-5\right)\left(x-3\right), the least common multiple of x-3,x-5,3.

$$\left(3x-15\right)\left(x-2\right)-\left(3x-9\right)\left(x-4\right)=10\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply 3x-15 by x-2 and combine like terms.

$$3x^{2}-21x+30-\left(3x-9\right)\left(x-4\right)=10\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply 3x-9 by x-4 and combine like terms.

$$3x^{2}-21x+30-\left(3x^{2}-21x+36\right)=10\left(x-5\right)\left(x-3\right)$$

To find the opposite of 3x^{2}-21x+36, find the opposite of each term.

$$3x^{2}-21x+30-3x^{2}+21x-36=10\left(x-5\right)\left(x-3\right)$$

Combine 3x^{2} and -3x^{2} to get 0.

$$-21x+30+21x-36=10\left(x-5\right)\left(x-3\right)$$

Combine -21x and 21x to get 0.

$$30-36=10\left(x-5\right)\left(x-3\right)$$

Subtract 36 from 30 to get -6.

$$-6=10\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply 10 by x-5.

$$-6=\left(10x-50\right)\left(x-3\right)$$

Use the distributive property to multiply 10x-50 by x-3 and combine like terms.

$$-6=10x^{2}-80x+150$$

Swap sides so that all variable terms are on the left hand side.

$$10x^{2}-80x+150=-6$$

Add 6 to both sides.

$$10x^{2}-80x+150+6=0$$

Add 150 and 6 to get 156.

$$10x^{2}-80x+156=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -80 for b, and 156 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 10\times 156}}{2\times 10}$$

Square -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 10\times 156}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-\left(-80\right)±\sqrt{6400-40\times 156}}{2\times 10}$$

Multiply -40 times 156.

$$x=\frac{-\left(-80\right)±\sqrt{6400-6240}}{2\times 10}$$

Add 6400 to -6240.

$$x=\frac{-\left(-80\right)±\sqrt{160}}{2\times 10}$$

Take the square root of 160.

$$x=\frac{-\left(-80\right)±4\sqrt{10}}{2\times 10}$$

The opposite of -80 is 80.

$$x=\frac{80±4\sqrt{10}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{80±4\sqrt{10}}{20}$$

Now solve the equation x=\frac{80±4\sqrt{10}}{20} when ± is plus. Add 80 to 4\sqrt{10}.

$$x=\frac{4\sqrt{10}+80}{20}$$

Divide 80+4\sqrt{10} by 20.

$$x=\frac{\sqrt{10}}{5}+4$$

Now solve the equation x=\frac{80±4\sqrt{10}}{20} when ± is minus. Subtract 4\sqrt{10} from 80.

$$x=\frac{80-4\sqrt{10}}{20}$$

Divide 80-4\sqrt{10} by 20.

$$x=-\frac{\sqrt{10}}{5}+4$$

The equation is now solved.

$$x=\frac{\sqrt{10}}{5}+4 x=-\frac{\sqrt{10}}{5}+4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 3,5 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-5\right)\left(x-3\right), the least common multiple of x-3,x-5,3.

$$\left(3x-15\right)\left(x-2\right)-\left(3x-9\right)\left(x-4\right)=10\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply 3x-15 by x-2 and combine like terms.

$$3x^{2}-21x+30-\left(3x-9\right)\left(x-4\right)=10\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply 3x-9 by x-4 and combine like terms.

$$3x^{2}-21x+30-\left(3x^{2}-21x+36\right)=10\left(x-5\right)\left(x-3\right)$$

To find the opposite of 3x^{2}-21x+36, find the opposite of each term.

$$3x^{2}-21x+30-3x^{2}+21x-36=10\left(x-5\right)\left(x-3\right)$$

Combine 3x^{2} and -3x^{2} to get 0.

$$-21x+30+21x-36=10\left(x-5\right)\left(x-3\right)$$

Combine -21x and 21x to get 0.

$$30-36=10\left(x-5\right)\left(x-3\right)$$

Subtract 36 from 30 to get -6.

$$-6=10\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply 10 by x-5.

$$-6=\left(10x-50\right)\left(x-3\right)$$

Use the distributive property to multiply 10x-50 by x-3 and combine like terms.

$$-6=10x^{2}-80x+150$$

Swap sides so that all variable terms are on the left hand side.

$$10x^{2}-80x+150=-6$$

Subtract 150 from both sides.

$$10x^{2}-80x=-6-150$$

Subtract 150 from -6 to get -156.

$$10x^{2}-80x=-156$$

Divide both sides by 10.

$$\frac{10x^{2}-80x}{10}=-\frac{156}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\left(-\frac{80}{10}\right)x=-\frac{156}{10}$$

Divide -80 by 10.

$$x^{2}-8x=-\frac{156}{10}$$

Reduce the fraction \frac{-156}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}-8x=-\frac{78}{5}$$

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

$$x^{2}-8x+\left(-4\right)^{2}=-\frac{78}{5}+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-\frac{78}{5}+16$$

Add -\frac{78}{5} to 16.

$$x^{2}-8x+16=\frac{2}{5}$$

Factor x^{2}-8x+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}.

$$\left(x-4\right)^{2}=\frac{2}{5}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{\frac{2}{5}}$$

Simplify.

$$x-4=\frac{\sqrt{10}}{5} x-4=-\frac{\sqrt{10}}{5}$$

Add 4 to both sides of the equation.

$$x=\frac{\sqrt{10}}{5}+4 x=-\frac{\sqrt{10}}{5}+4$$