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

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

Solve for x

$$x = \frac{\sqrt{401} + 9}{2} \approx 14.512492197$$

$$x=\frac{9-\sqrt{401}}{2}\approx -5.512492197$$

Steps Using the Quadratic Formula

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

$$\left(2x+12\right)\times 5+\left(2x+10\right)\times 5=\left(x+5\right)\left(x+6\right)$$

Use the distributive property to multiply 2x+12 by 5.

$$10x+60+\left(2x+10\right)\times 5=\left(x+5\right)\left(x+6\right)$$

Use the distributive property to multiply 2x+10 by 5.

$$10x+60+10x+50=\left(x+5\right)\left(x+6\right)$$

Combine 10x and 10x to get 20x.

$$20x+60+50=\left(x+5\right)\left(x+6\right)$$

Add 60 and 50 to get 110.

$$20x+110=\left(x+5\right)\left(x+6\right)$$

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

$$20x+110=x^{2}+11x+30$$

Subtract x^{2} from both sides.

$$20x+110-x^{2}=11x+30$$

Subtract 11x from both sides.

$$20x+110-x^{2}-11x=30$$

Combine 20x and -11x to get 9x.

$$9x+110-x^{2}=30$$

Subtract 30 from both sides.

$$9x+110-x^{2}-30=0$$

Subtract 30 from 110 to get 80.

$$9x+80-x^{2}=0$$

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.

$$-x^{2}+9x+80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and 80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\times 80}}{2\left(-1\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\times 80}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-9±\sqrt{81+4\times 80}}{2\left(-1\right)}$$

Multiply 4 times 80.

$$x=\frac{-9±\sqrt{81+320}}{2\left(-1\right)}$$

Add 81 to 320.

$$x=\frac{-9±\sqrt{401}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±\sqrt{401}}{-2}$$

Now solve the equation x=\frac{-9±\sqrt{401}}{-2} when ± is plus. Add -9 to \sqrt{401}.

$$x=\frac{\sqrt{401}-9}{-2}$$

Divide -9+\sqrt{401} by -2.

$$x=\frac{9-\sqrt{401}}{2}$$

Now solve the equation x=\frac{-9±\sqrt{401}}{-2} when ± is minus. Subtract \sqrt{401} from -9.

$$x=\frac{-\sqrt{401}-9}{-2}$$

Divide -9-\sqrt{401} by -2.

$$x=\frac{\sqrt{401}+9}{2}$$

The equation is now solved.

$$x=\frac{9-\sqrt{401}}{2} x=\frac{\sqrt{401}+9}{2}$$

Steps for Completing the Square

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

$$\left(2x+12\right)\times 5+\left(2x+10\right)\times 5=\left(x+5\right)\left(x+6\right)$$

Use the distributive property to multiply 2x+12 by 5.

$$10x+60+\left(2x+10\right)\times 5=\left(x+5\right)\left(x+6\right)$$

Use the distributive property to multiply 2x+10 by 5.

$$10x+60+10x+50=\left(x+5\right)\left(x+6\right)$$

Combine 10x and 10x to get 20x.

$$20x+60+50=\left(x+5\right)\left(x+6\right)$$

Add 60 and 50 to get 110.

$$20x+110=\left(x+5\right)\left(x+6\right)$$

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

$$20x+110=x^{2}+11x+30$$

Subtract x^{2} from both sides.

$$20x+110-x^{2}=11x+30$$

Subtract 11x from both sides.

$$20x+110-x^{2}-11x=30$$

Combine 20x and -11x to get 9x.

$$9x+110-x^{2}=30$$

Subtract 110 from both sides.

$$9x-x^{2}=30-110$$

Subtract 110 from 30 to get -80.

$$9x-x^{2}=-80$$

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.

$$-x^{2}+9x=-80$$

Divide both sides by -1.

$$\frac{-x^{2}+9x}{-1}=-\frac{80}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{9}{-1}x=-\frac{80}{-1}$$

Divide 9 by -1.

$$x^{2}-9x=-\frac{80}{-1}$$

Divide -80 by -1.

$$x^{2}-9x=80$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=80+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=80+\frac{81}{4}$$

Add 80 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{401}{4}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{401}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{401}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{401}}{2} x-\frac{9}{2}=-\frac{\sqrt{401}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{401}+9}{2} x=\frac{9-\sqrt{401}}{2}$$