Solve 9/9+x=4/4+x+10%

$$\frac { 9 } { 9 + x } = \frac { 4 } { 4 + x } + 10 \%$$

Solve for x

$$x=1$$

$$x=36$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -9,-4 since division by zero is not defined. Multiply both sides of the equation by 100\left(x+4\right)\left(x+9\right), the least common multiple of 9+x,4+x,100.

$$\left(100x+400\right)\times 9=\left(100x+900\right)\times 4+\left(x+4\right)\left(x+9\right)\times 10$$

Use the distributive property to multiply 100x+400 by 9.

$$900x+3600=\left(100x+900\right)\times 4+\left(x+4\right)\left(x+9\right)\times 10$$

Use the distributive property to multiply 100x+900 by 4.

$$900x+3600=400x+3600+\left(x+4\right)\left(x+9\right)\times 10$$

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

$$900x+3600=400x+3600+\left(x^{2}+13x+36\right)\times 10$$

Use the distributive property to multiply x^{2}+13x+36 by 10.

$$900x+3600=400x+3600+10x^{2}+130x+360$$

Combine 400x and 130x to get 530x.

$$900x+3600=530x+3600+10x^{2}+360$$

Add 3600 and 360 to get 3960.

$$900x+3600=530x+3960+10x^{2}$$

Subtract 530x from both sides.

$$900x+3600-530x=3960+10x^{2}$$

Combine 900x and -530x to get 370x.

$$370x+3600=3960+10x^{2}$$

Subtract 3960 from both sides.

$$370x+3600-3960=10x^{2}$$

Subtract 3960 from 3600 to get -360.

$$370x-360=10x^{2}$$

Subtract 10x^{2} from both sides.

$$370x-360-10x^{2}=0$$

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.

$$-10x^{2}+370x-360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 370 for b, and -360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-370±\sqrt{370^{2}-4\left(-10\right)\left(-360\right)}}{2\left(-10\right)}$$

Square 370.

$$x=\frac{-370±\sqrt{136900-4\left(-10\right)\left(-360\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-370±\sqrt{136900+40\left(-360\right)}}{2\left(-10\right)}$$

Multiply 40 times -360.

$$x=\frac{-370±\sqrt{136900-14400}}{2\left(-10\right)}$$

Add 136900 to -14400.

$$x=\frac{-370±\sqrt{122500}}{2\left(-10\right)}$$

Take the square root of 122500.

$$x=\frac{-370±350}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-370±350}{-20}$$

Now solve the equation x=\frac{-370±350}{-20} when ± is plus. Add -370 to 350.

$$x=-\frac{20}{-20}$$

Divide -20 by -20.

$$x=1$$

Now solve the equation x=\frac{-370±350}{-20} when ± is minus. Subtract 350 from -370.

$$x=-\frac{720}{-20}$$

Divide -720 by -20.

$$x=36$$

The equation is now solved.

$$x=1 x=36$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -9,-4 since division by zero is not defined. Multiply both sides of the equation by 100\left(x+4\right)\left(x+9\right), the least common multiple of 9+x,4+x,100.

$$\left(100x+400\right)\times 9=\left(100x+900\right)\times 4+\left(x+4\right)\left(x+9\right)\times 10$$

Use the distributive property to multiply 100x+400 by 9.

$$900x+3600=\left(100x+900\right)\times 4+\left(x+4\right)\left(x+9\right)\times 10$$

Use the distributive property to multiply 100x+900 by 4.

$$900x+3600=400x+3600+\left(x+4\right)\left(x+9\right)\times 10$$

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

$$900x+3600=400x+3600+\left(x^{2}+13x+36\right)\times 10$$

Use the distributive property to multiply x^{2}+13x+36 by 10.

$$900x+3600=400x+3600+10x^{2}+130x+360$$

Combine 400x and 130x to get 530x.

$$900x+3600=530x+3600+10x^{2}+360$$

Add 3600 and 360 to get 3960.

$$900x+3600=530x+3960+10x^{2}$$

Subtract 530x from both sides.

$$900x+3600-530x=3960+10x^{2}$$

Combine 900x and -530x to get 370x.

$$370x+3600=3960+10x^{2}$$

Subtract 10x^{2} from both sides.

$$370x+3600-10x^{2}=3960$$

Subtract 3600 from both sides.

$$370x-10x^{2}=3960-3600$$

Subtract 3600 from 3960 to get 360.

$$370x-10x^{2}=360$$

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.

$$-10x^{2}+370x=360$$

Divide both sides by -10.

$$\frac{-10x^{2}+370x}{-10}=\frac{360}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{370}{-10}x=\frac{360}{-10}$$

Divide 370 by -10.

$$x^{2}-37x=\frac{360}{-10}$$

Divide 360 by -10.

$$x^{2}-37x=-36$$

Divide -37, the coefficient of the x term, by 2 to get -\frac{37}{2}. Then add the square of -\frac{37}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-37x+\left(-\frac{37}{2}\right)^{2}=-36+\left(-\frac{37}{2}\right)^{2}$$

Square -\frac{37}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-37x+\frac{1369}{4}=-36+\frac{1369}{4}$$

Add -36 to \frac{1369}{4}.

$$x^{2}-37x+\frac{1369}{4}=\frac{1225}{4}$$

Factor x^{2}-37x+\frac{1369}{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}.

$$\left(x-\frac{37}{2}\right)^{2}=\frac{1225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{37}{2}\right)^{2}}=\sqrt{\frac{1225}{4}}$$

Simplify.

$$x-\frac{37}{2}=\frac{35}{2} x-\frac{37}{2}=-\frac{35}{2}$$

Add \frac{37}{2} to both sides of the equation.

$$x=36 x=1$$