Solve 11+x/35+11/35+2x=1/3

$$\frac{ 11+x }{ 35 } + \frac{ 11 }{ 35+2x } = \frac{ 1 }{ 3 }$$

Solve for x (complex solution)

$$x=\frac{-101+\sqrt{15839}i}{12}\approx -8.416666667+10.487757413i$$

$$x=\frac{-\sqrt{15839}i-101}{12}\approx -8.416666667-10.487757413i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -\frac{35}{2} since division by zero is not defined. Multiply both sides of the equation by 105\left(2x+35\right), the least common multiple of 35,35+2x,3.

$$\left(6x+105\right)\left(11+x\right)+105\times 11=35\left(2x+35\right)$$

Use the distributive property to multiply 6x+105 by 11+x and combine like terms.

$$171x+6x^{2}+1155+105\times 11=35\left(2x+35\right)$$

Multiply 105 and 11 to get 1155.

$$171x+6x^{2}+1155+1155=35\left(2x+35\right)$$

Add 1155 and 1155 to get 2310.

$$171x+6x^{2}+2310=35\left(2x+35\right)$$

Use the distributive property to multiply 35 by 2x+35.

$$171x+6x^{2}+2310=70x+1225$$

Subtract 70x from both sides.

$$171x+6x^{2}+2310-70x=1225$$

Combine 171x and -70x to get 101x.

$$101x+6x^{2}+2310=1225$$

Subtract 1225 from both sides.

$$101x+6x^{2}[phone]=0$$

Subtract 1225 from 2310 to get 1085.

$$101x+6x^{2}+1085=0$$

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.

$$6x^{2}+101x+1085=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 101 for b, and 1085 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-101±\sqrt{101^{2}-4\times 6\times 1085}}{2\times 6}$$

Square 101.

$$x=\frac{-101±\sqrt{10201-4\times 6\times 1085}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-101±\sqrt{10201-24\times 1085}}{2\times 6}$$

Multiply -24 times 1085.

$$x=\frac{-101±\sqrt{10201-26040}}{2\times 6}$$

Add 10201 to -26040.

$$x=\frac{-101±\sqrt{-15839}}{2\times 6}$$

Take the square root of -15839.

$$x=\frac{-101±\sqrt{15839}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-101±\sqrt{15839}i}{12}$$

Now solve the equation x=\frac{-101±\sqrt{15839}i}{12} when ± is plus. Add -101 to i\sqrt{15839}.

$$x=\frac{-101+\sqrt{15839}i}{12}$$

Now solve the equation x=\frac{-101±\sqrt{15839}i}{12} when ± is minus. Subtract i\sqrt{15839} from -101.

$$x=\frac{-\sqrt{15839}i-101}{12}$$

The equation is now solved.

$$x=\frac{-101+\sqrt{15839}i}{12} x=\frac{-\sqrt{15839}i-101}{12}$$

Steps for Completing the Square

Variable x cannot be equal to -\frac{35}{2} since division by zero is not defined. Multiply both sides of the equation by 105\left(2x+35\right), the least common multiple of 35,35+2x,3.

$$\left(6x+105\right)\left(11+x\right)+105\times 11=35\left(2x+35\right)$$

Use the distributive property to multiply 6x+105 by 11+x and combine like terms.

$$171x+6x^{2}+1155+105\times 11=35\left(2x+35\right)$$

Multiply 105 and 11 to get 1155.

$$171x+6x^{2}+1155+1155=35\left(2x+35\right)$$

Add 1155 and 1155 to get 2310.

$$171x+6x^{2}+2310=35\left(2x+35\right)$$

Use the distributive property to multiply 35 by 2x+35.

$$171x+6x^{2}+2310=70x+1225$$

Subtract 70x from both sides.

$$171x+6x^{2}+2310-70x=1225$$

Combine 171x and -70x to get 101x.

$$101x+6x^{2}+2310=1225$$

Subtract 2310 from both sides.

$$101x+6x^{2}=1225-2310$$

Subtract 2310 from 1225 to get -1085.

$$101x+6x^{2}=-1085$$

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.

$$6x^{2}+101x=-1085$$

Divide both sides by 6.

$$\frac{6x^{2}+101x}{6}=-\frac{1085}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{101}{6}x=-\frac{1085}{6}$$

Divide \frac{101}{6}, the coefficient of the x term, by 2 to get \frac{101}{12}. Then add the square of \frac{101}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{101}{6}x+\left(\frac{101}{12}\right)^{2}=-\frac{1085}{6}+\left(\frac{101}{12}\right)^{2}$$

Square \frac{101}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{101}{6}x+\frac{10201}{144}=-\frac{1085}{6}+\frac{10201}{144}$$

Add -\frac{1085}{6} to \frac{10201}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{101}{6}x+\frac{10201}{144}=-\frac{15839}{144}$$

Factor x^{2}+\frac{101}{6}x+\frac{10201}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{101}{12}\right)^{2}=-\frac{15839}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{101}{12}\right)^{2}}=\sqrt{-\frac{15839}{144}}$$

Simplify.

$$x+\frac{101}{12}=\frac{\sqrt{15839}i}{12} x+\frac{101}{12}=-\frac{\sqrt{15839}i}{12}$$

Subtract \frac{101}{12} from both sides of the equation.

$$x=\frac{-101+\sqrt{15839}i}{12} x=\frac{-\sqrt{15839}i-101}{12}$$