Solve 6left(1+xright)+6left(1+xright)left(1+xright)=17.34-6

$$6 \left( 1+x \right) +6 \left( 1+x \right) \left( 1+x \right) = 17.34-6$$

Solve for x

$$x=\frac{\sqrt{214}}{10}-1.5\approx -0.037126116$$

$$x=-\frac{\sqrt{214}}{10}-1.5\approx -2.962873884$$

Steps Using the Quadratic Formula

Multiply 1+x and 1+x to get \left(1+x\right)^{2}.

$$6\left(1+x\right)+6\left(1+x\right)^{2}=17.34-6$$

Use the distributive property to multiply 6 by 1+x.

$$6+6x+6\left(1+x\right)^{2}=17.34-6$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$6+6x+6\left(1+2x+x^{2}\right)=17.34-6$$

Use the distributive property to multiply 6 by 1+2x+x^{2}.

$$6+6x+6+12x+6x^{2}=17.34-6$$

Add 6 and 6 to get 12.

$$12+6x+12x+6x^{2}=17.34-6$$

Combine 6x and 12x to get 18x.

$$12+18x+6x^{2}=17.34-6$$

Subtract 6 from 17.34 to get 11.34.

$$12+18x+6x^{2}=11.34$$

Subtract 11.34 from both sides.

$$12+18x+6x^{2}-11.34=0$$

Subtract 11.34 from 12 to get 0.66.

$$0.66+18x+6x^{2}=0$$

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.

$$6x^{2}+18x+0.66=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 18 for b, and 0.66 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 6\times 0.66}}{2\times 6}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 6\times 0.66}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-18±\sqrt{324-24\times 0.66}}{2\times 6}$$

Multiply -24 times 0.66.

$$x=\frac{-18±\sqrt{324-15.84}}{2\times 6}$$

Add 324 to -15.84.

$$x=\frac{-18±\sqrt{308.16}}{2\times 6}$$

Take the square root of 308.16.

$$x=\frac{-18±\frac{6\sqrt{214}}{5}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-18±\frac{6\sqrt{214}}{5}}{12}$$

Now solve the equation x=\frac{-18±\frac{6\sqrt{214}}{5}}{12} when ± is plus. Add -18 to \frac{6\sqrt{214}}{5}.

$$x=\frac{\frac{6\sqrt{214}}{5}-18}{12}$$

Divide -18+\frac{6\sqrt{214}}{5} by 12.

$$x=\frac{\sqrt{214}}{10}-\frac{3}{2}$$

Now solve the equation x=\frac{-18±\frac{6\sqrt{214}}{5}}{12} when ± is minus. Subtract \frac{6\sqrt{214}}{5} from -18.

$$x=\frac{-\frac{6\sqrt{214}}{5}-18}{12}$$

Divide -18-\frac{6\sqrt{214}}{5} by 12.

$$x=-\frac{\sqrt{214}}{10}-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{214}}{10}-\frac{3}{2} x=-\frac{\sqrt{214}}{10}-\frac{3}{2}$$

Steps for Completing the Square

Multiply 1+x and 1+x to get \left(1+x\right)^{2}.

$$6\left(1+x\right)+6\left(1+x\right)^{2}=17.34-6$$

Use the distributive property to multiply 6 by 1+x.

$$6+6x+6\left(1+x\right)^{2}=17.34-6$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+x\right)^{2}.

$$6+6x+6\left(1+2x+x^{2}\right)=17.34-6$$

Use the distributive property to multiply 6 by 1+2x+x^{2}.

$$6+6x+6+12x+6x^{2}=17.34-6$$

Add 6 and 6 to get 12.

$$12+6x+12x+6x^{2}=17.34-6$$

Combine 6x and 12x to get 18x.

$$12+18x+6x^{2}=17.34-6$$

Subtract 6 from 17.34 to get 11.34.

$$12+18x+6x^{2}=11.34$$

Subtract 12 from both sides.

$$18x+6x^{2}=11.34-12$$

Subtract 12 from 11.34 to get -0.66.

$$18x+6x^{2}=-0.66$$

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.

$$6x^{2}+18x=-0.66$$

Divide both sides by 6.

$$\frac{6x^{2}+18x}{6}=-\frac{0.66}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{18}{6}x=-\frac{0.66}{6}$$

Divide 18 by 6.

$$x^{2}+3x=-\frac{0.66}{6}$$

Divide -0.66 by 6.

$$x^{2}+3x=-0.11$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-0.11+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=-0.11+\frac{9}{4}$$

Add -0.11 to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=\frac{107}{50}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{107}{50}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{107}{50}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{214}}{10} x+\frac{3}{2}=-\frac{\sqrt{214}}{10}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{214}}{10}-\frac{3}{2} x=-\frac{\sqrt{214}}{10}-\frac{3}{2}$$