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

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

Solve for x (complex solution)

$$x=\sqrt{10}-1\approx 2.16227766$$

$$x=-\left(\sqrt{10}+1\right)\approx -4.16227766$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+3\right)\left(x+5\right), the least common multiple of 2x+6,x+5.

$$\left(x+5\right)\times 6+\left(2x+6\right)\times 3=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply x+5 by 6.

$$6x+30+\left(2x+6\right)\times 3=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2x+6 by 3.

$$6x+30+6x+18=2\left(x+3\right)\left(x+5\right)$$

Combine 6x and 6x to get 12x.

$$12x+30+18=2\left(x+3\right)\left(x+5\right)$$

Add 30 and 18 to get 48.

$$12x+48=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x+3.

$$12x+48=\left(2x+6\right)\left(x+5\right)$$

Use the distributive property to multiply 2x+6 by x+5 and combine like terms.

$$12x+48=2x^{2}+16x+30$$

Subtract 2x^{2} from both sides.

$$12x+48-2x^{2}=16x+30$$

Subtract 16x from both sides.

$$12x+48-2x^{2}-16x=30$$

Combine 12x and -16x to get -4x.

$$-4x+48-2x^{2}=30$$

Subtract 30 from both sides.

$$-4x+48-2x^{2}-30=0$$

Subtract 30 from 48 to get 18.

$$-4x+18-2x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-2x^{2}-4x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -4 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-4\right)±\sqrt{16+8\times 18}}{2\left(-2\right)}$$

Multiply 8 times 18.

$$x=\frac{-\left(-4\right)±\sqrt{16+144}}{2\left(-2\right)}$$

Add 16 to 144.

$$x=\frac{-\left(-4\right)±\sqrt{160}}{2\left(-2\right)}$$

Take the square root of 160.

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

The opposite of -4 is 4.

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

Multiply 2 times -2.

$$x=\frac{4±4\sqrt{10}}{-4}$$

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

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

Divide 4+4\sqrt{10} by -4.

$$x=-\left(\sqrt{10}+1\right)$$

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

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

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

$$x=\sqrt{10}-1$$

The equation is now solved.

$$x=-\left(\sqrt{10}+1\right) x=\sqrt{10}-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+3\right)\left(x+5\right), the least common multiple of 2x+6,x+5.

$$\left(x+5\right)\times 6+\left(2x+6\right)\times 3=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply x+5 by 6.

$$6x+30+\left(2x+6\right)\times 3=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2x+6 by 3.

$$6x+30+6x+18=2\left(x+3\right)\left(x+5\right)$$

Combine 6x and 6x to get 12x.

$$12x+30+18=2\left(x+3\right)\left(x+5\right)$$

Add 30 and 18 to get 48.

$$12x+48=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x+3.

$$12x+48=\left(2x+6\right)\left(x+5\right)$$

Use the distributive property to multiply 2x+6 by x+5 and combine like terms.

$$12x+48=2x^{2}+16x+30$$

Subtract 2x^{2} from both sides.

$$12x+48-2x^{2}=16x+30$$

Subtract 16x from both sides.

$$12x+48-2x^{2}-16x=30$$

Combine 12x and -16x to get -4x.

$$-4x+48-2x^{2}=30$$

Subtract 48 from both sides.

$$-4x-2x^{2}=30-48$$

Subtract 48 from 30 to get -18.

$$-4x-2x^{2}=-18$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-2x^{2}-4x=-18$$

Divide both sides by -2.

$$\frac{-2x^{2}-4x}{-2}=-\frac{18}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{4}{-2}\right)x=-\frac{18}{-2}$$

Divide -4 by -2.

$$x^{2}+2x=-\frac{18}{-2}$$

Divide -18 by -2.

$$x^{2}+2x=9$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=9+1^{2}$$

Square 1.

$$x^{2}+2x+1=9+1$$

Add 9 to 1.

$$x^{2}+2x+1=10$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{10}$$

Simplify.

$$x+1=\sqrt{10} x+1=-\sqrt{10}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{10}-1 x=-\sqrt{10}-1$$

Solve for x

$$x=\sqrt{10}-1\approx 2.16227766$$

$$x=-\sqrt{10}-1\approx -4.16227766$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+3\right)\left(x+5\right), the least common multiple of 2x+6,x+5.

$$\left(x+5\right)\times 6+\left(2x+6\right)\times 3=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply x+5 by 6.

$$6x+30+\left(2x+6\right)\times 3=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2x+6 by 3.

$$6x+30+6x+18=2\left(x+3\right)\left(x+5\right)$$

Combine 6x and 6x to get 12x.

$$12x+30+18=2\left(x+3\right)\left(x+5\right)$$

Add 30 and 18 to get 48.

$$12x+48=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x+3.

$$12x+48=\left(2x+6\right)\left(x+5\right)$$

Use the distributive property to multiply 2x+6 by x+5 and combine like terms.

$$12x+48=2x^{2}+16x+30$$

Subtract 2x^{2} from both sides.

$$12x+48-2x^{2}=16x+30$$

Subtract 16x from both sides.

$$12x+48-2x^{2}-16x=30$$

Combine 12x and -16x to get -4x.

$$-4x+48-2x^{2}=30$$

Subtract 30 from both sides.

$$-4x+48-2x^{2}-30=0$$

Subtract 30 from 48 to get 18.

$$-4x+18-2x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-2x^{2}-4x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -4 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-4\right)±\sqrt{16+8\times 18}}{2\left(-2\right)}$$

Multiply 8 times 18.

$$x=\frac{-\left(-4\right)±\sqrt{16+144}}{2\left(-2\right)}$$

Add 16 to 144.

$$x=\frac{-\left(-4\right)±\sqrt{160}}{2\left(-2\right)}$$

Take the square root of 160.

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

The opposite of -4 is 4.

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

Multiply 2 times -2.

$$x=\frac{4±4\sqrt{10}}{-4}$$

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

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

Divide 4+4\sqrt{10} by -4.

$$x=-\left(\sqrt{10}+1\right)$$

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

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

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

$$x=\sqrt{10}-1$$

The equation is now solved.

$$x=-\left(\sqrt{10}+1\right) x=\sqrt{10}-1$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,-3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x+3\right)\left(x+5\right), the least common multiple of 2x+6,x+5.

$$\left(x+5\right)\times 6+\left(2x+6\right)\times 3=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply x+5 by 6.

$$6x+30+\left(2x+6\right)\times 3=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2x+6 by 3.

$$6x+30+6x+18=2\left(x+3\right)\left(x+5\right)$$

Combine 6x and 6x to get 12x.

$$12x+30+18=2\left(x+3\right)\left(x+5\right)$$

Add 30 and 18 to get 48.

$$12x+48=2\left(x+3\right)\left(x+5\right)$$

Use the distributive property to multiply 2 by x+3.

$$12x+48=\left(2x+6\right)\left(x+5\right)$$

Use the distributive property to multiply 2x+6 by x+5 and combine like terms.

$$12x+48=2x^{2}+16x+30$$

Subtract 2x^{2} from both sides.

$$12x+48-2x^{2}=16x+30$$

Subtract 16x from both sides.

$$12x+48-2x^{2}-16x=30$$

Combine 12x and -16x to get -4x.

$$-4x+48-2x^{2}=30$$

Subtract 48 from both sides.

$$-4x-2x^{2}=30-48$$

Subtract 48 from 30 to get -18.

$$-4x-2x^{2}=-18$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-2x^{2}-4x=-18$$

Divide both sides by -2.

$$\frac{-2x^{2}-4x}{-2}=-\frac{18}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{4}{-2}\right)x=-\frac{18}{-2}$$

Divide -4 by -2.

$$x^{2}+2x=-\frac{18}{-2}$$

Divide -18 by -2.

$$x^{2}+2x=9$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=9+1^{2}$$

Square 1.

$$x^{2}+2x+1=9+1$$

Add 9 to 1.

$$x^{2}+2x+1=10$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{10}$$

Simplify.

$$x+1=\sqrt{10} x+1=-\sqrt{10}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{10}-1 x=-\sqrt{10}-1$$