Solve 10/3+5-x/x+5=x+5/x-5

$$\frac { 10 } { 3 } + \frac { 5 - x } { x + 5 } = \frac { x + 5 } { x - 5 }$$

Solve for x

$$x=-10$$

$$x=10$$

Steps by Finding Square Root

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

$$3\left(x-5\right)\left(x+5\right)\times \frac{10}{3}+\left(3x-15\right)\left(5-x\right)=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3 by x-5.

$$\left(3x-15\right)\left(x+5\right)\times \frac{10}{3}+\left(3x-15\right)\left(5-x\right)=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3x-15 by x+5 and combine like terms.

$$\left(3x^{2}-75\right)\times \frac{10}{3}+\left(3x-15\right)\left(5-x\right)=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3x^{2}-75 by \frac{10}{3}.

$$10x^{2}-250+\left(3x-15\right)\left(5-x\right)=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3x-15 by 5-x and combine like terms.

$$10x^{2}-250+30x-3x^{2}-75=\left(3x+15\right)\left(x+5\right)$$

Combine 10x^{2} and -3x^{2} to get 7x^{2}.

$$7x^{2}-250+30x-75=\left(3x+15\right)\left(x+5\right)$$

Subtract 75 from -250 to get -325.

$$7x^{2}-325+30x=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3x+15 by x+5 and combine like terms.

$$7x^{2}-325+30x=3x^{2}+30x+75$$

Subtract 3x^{2} from both sides.

$$7x^{2}-325+30x-3x^{2}=30x+75$$

Combine 7x^{2} and -3x^{2} to get 4x^{2}.

$$4x^{2}-325+30x=30x+75$$

Subtract 30x from both sides.

$$4x^{2}-325+30x-30x=75$$

Combine 30x and -30x to get 0.

$$4x^{2}-325=75$$

Add 325 to both sides.

$$4x^{2}=75+325$$

Add 75 and 325 to get 400.

$$4x^{2}=400$$

Divide both sides by 4.

$$x^{2}=\frac{400}{4}$$

Divide 400 by 4 to get 100.

$$x^{2}=100$$

Take the square root of both sides of the equation.

$$x=10 x=-10$$

Steps Using the Quadratic Formula

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

$$3\left(x-5\right)\left(x+5\right)\times \frac{10}{3}+\left(3x-15\right)\left(5-x\right)=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3 by x-5.

$$\left(3x-15\right)\left(x+5\right)\times \frac{10}{3}+\left(3x-15\right)\left(5-x\right)=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3x-15 by x+5 and combine like terms.

$$\left(3x^{2}-75\right)\times \frac{10}{3}+\left(3x-15\right)\left(5-x\right)=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3x^{2}-75 by \frac{10}{3}.

$$10x^{2}-250+\left(3x-15\right)\left(5-x\right)=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3x-15 by 5-x and combine like terms.

$$10x^{2}-250+30x-3x^{2}-75=\left(3x+15\right)\left(x+5\right)$$

Combine 10x^{2} and -3x^{2} to get 7x^{2}.

$$7x^{2}-250+30x-75=\left(3x+15\right)\left(x+5\right)$$

Subtract 75 from -250 to get -325.

$$7x^{2}-325+30x=\left(3x+15\right)\left(x+5\right)$$

Use the distributive property to multiply 3x+15 by x+5 and combine like terms.

$$7x^{2}-325+30x=3x^{2}+30x+75$$

Subtract 3x^{2} from both sides.

$$7x^{2}-325+30x-3x^{2}=30x+75$$

Combine 7x^{2} and -3x^{2} to get 4x^{2}.

$$4x^{2}-325+30x=30x+75$$

Subtract 30x from both sides.

$$4x^{2}-325+30x-30x=75$$

Combine 30x and -30x to get 0.

$$4x^{2}-325=75$$

Subtract 75 from both sides.

$$4x^{2}-325-75=0$$

Subtract 75 from -325 to get -400.

$$4x^{2}-400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 0 for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 4\left(-400\right)}}{2\times 4}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 4\left(-400\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{0±\sqrt{-16\left(-400\right)}}{2\times 4}$$

Multiply -16 times -400.

$$x=\frac{0±\sqrt{6400}}{2\times 4}$$

Take the square root of 6400.

$$x=\frac{0±80}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{0±80}{8}$$

Now solve the equation x=\frac{0±80}{8} when ± is plus. Divide 80 by 8.

$$x=10$$

Now solve the equation x=\frac{0±80}{8} when ± is minus. Divide -80 by 8.

$$x=-10$$

The equation is now solved.

$$x=10 x=-10$$