Solve 35=6(11+x)(1+x)

$$35 = 6 ( 11 + x ) ( 1 + x )$$

Solve for x

$$x=\frac{\sqrt{1110}}{6}-6\approx -0.447222917$$

$$x=-\frac{\sqrt{1110}}{6}-6\approx -11.552777083$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6 by 11+x.

$$35=\left(66+6x\right)\left(1+x\right)$$

Use the distributive property to multiply 66+6x by 1+x and combine like terms.

$$35=66+72x+6x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$66+72x+6x^{2}=35$$

Subtract 35 from both sides.

$$66+72x+6x^{2}-35=0$$

Subtract 35 from 66 to get 31.

$$31+72x+6x^{2}=0$$

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.

$$6x^{2}+72x+31=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 72 for b, and 31 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-72±\sqrt{72^{2}-4\times 6\times 31}}{2\times 6}$$

Square 72.

$$x=\frac{-72±\sqrt{5184-4\times 6\times 31}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-72±\sqrt{5184-24\times 31}}{2\times 6}$$

Multiply -24 times 31.

$$x=\frac{-72±\sqrt{5184-744}}{2\times 6}$$

Add 5184 to -744.

$$x=\frac{-72±\sqrt{4440}}{2\times 6}$$

Take the square root of 4440.

$$x=\frac{-72±2\sqrt{1110}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-72±2\sqrt{1110}}{12}$$

Now solve the equation x=\frac{-72±2\sqrt{1110}}{12} when ± is plus. Add -72 to 2\sqrt{1110}.

$$x=\frac{2\sqrt{1110}-72}{12}$$

Divide -72+2\sqrt{1110} by 12.

$$x=\frac{\sqrt{1110}}{6}-6$$

Now solve the equation x=\frac{-72±2\sqrt{1110}}{12} when ± is minus. Subtract 2\sqrt{1110} from -72.

$$x=\frac{-2\sqrt{1110}-72}{12}$$

Divide -72-2\sqrt{1110} by 12.

$$x=-\frac{\sqrt{1110}}{6}-6$$

The equation is now solved.

$$x=\frac{\sqrt{1110}}{6}-6 x=-\frac{\sqrt{1110}}{6}-6$$

Steps for Completing the Square

Use the distributive property to multiply 6 by 11+x.

$$35=\left(66+6x\right)\left(1+x\right)$$

Use the distributive property to multiply 66+6x by 1+x and combine like terms.

$$35=66+72x+6x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$66+72x+6x^{2}=35$$

Subtract 66 from both sides.

$$72x+6x^{2}=35-66$$

Subtract 66 from 35 to get -31.

$$72x+6x^{2}=-31$$

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.

$$6x^{2}+72x=-31$$

Divide both sides by 6.

$$\frac{6x^{2}+72x}{6}=-\frac{31}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{72}{6}x=-\frac{31}{6}$$

Divide 72 by 6.

$$x^{2}+12x=-\frac{31}{6}$$

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

$$x^{2}+12x+6^{2}=-\frac{31}{6}+6^{2}$$

Square 6.

$$x^{2}+12x+36=-\frac{31}{6}+36$$

Add -\frac{31}{6} to 36.

$$x^{2}+12x+36=\frac{185}{6}$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=\frac{185}{6}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{\frac{185}{6}}$$

Simplify.

$$x+6=\frac{\sqrt{1110}}{6} x+6=-\frac{\sqrt{1110}}{6}$$

Subtract 6 from both sides of the equation.

$$x=\frac{\sqrt{1110}}{6}-6 x=-\frac{\sqrt{1110}}{6}-6$$