Solve left(1-3xright)left(2+6xright)-left(-3x+1right)^2=1

$$\left( 1-3x \right) \left( 2+6x \right) - { \left(-3x+1 \right) }^{ 2 } =1$$

Solve for x

$$x=\frac{2}{9}\approx 0.222222222$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 1-3x by 2+6x and combine like terms.

$$2-18x^{2}-\left(-3x+1\right)^{2}=1$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-3x+1\right)^{2}.

$$2-18x^{2}-\left(9x^{2}-6x+1\right)=1$$

To find the opposite of 9x^{2}-6x+1, find the opposite of each term.

$$2-18x^{2}-9x^{2}+6x-1=1$$

Combine -18x^{2} and -9x^{2} to get -27x^{2}.

$$2-27x^{2}+6x-1=1$$

Subtract 1 from 2 to get 1.

$$1-27x^{2}+6x=1$$

Subtract 1 from both sides.

$$1-27x^{2}+6x-1=0$$

Subtract 1 from 1 to get 0.

$$-27x^{2}+6x=0$$

Factor out x.

$$x\left(-27x+6\right)=0$$

To find equation solutions, solve x=0 and -27x+6=0.

$$x=0 x=\frac{2}{9}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1-3x by 2+6x and combine like terms.

$$2-18x^{2}-\left(-3x+1\right)^{2}=1$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-3x+1\right)^{2}.

$$2-18x^{2}-\left(9x^{2}-6x+1\right)=1$$

To find the opposite of 9x^{2}-6x+1, find the opposite of each term.

$$2-18x^{2}-9x^{2}+6x-1=1$$

Combine -18x^{2} and -9x^{2} to get -27x^{2}.

$$2-27x^{2}+6x-1=1$$

Subtract 1 from 2 to get 1.

$$1-27x^{2}+6x=1$$

Subtract 1 from both sides.

$$1-27x^{2}+6x-1=0$$

Subtract 1 from 1 to get 0.

$$-27x^{2}+6x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -27 for a, 6 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}}}{2\left(-27\right)}$$

Take the square root of 6^{2}.

$$x=\frac{-6±6}{2\left(-27\right)}$$

Multiply 2 times -27.

$$x=\frac{-6±6}{-54}$$

Now solve the equation x=\frac{-6±6}{-54} when ± is plus. Add -6 to 6.

$$x=\frac{0}{-54}$$

Divide 0 by -54.

$$x=0$$

Now solve the equation x=\frac{-6±6}{-54} when ± is minus. Subtract 6 from -6.

$$x=-\frac{12}{-54}$$

Reduce the fraction \frac{-12}{-54} to lowest terms by extracting and canceling out 6.

$$x=\frac{2}{9}$$

The equation is now solved.

$$x=0 x=\frac{2}{9}$$

Steps for Completing the Square

Use the distributive property to multiply 1-3x by 2+6x and combine like terms.

$$2-18x^{2}-\left(-3x+1\right)^{2}=1$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-3x+1\right)^{2}.

$$2-18x^{2}-\left(9x^{2}-6x+1\right)=1$$

To find the opposite of 9x^{2}-6x+1, find the opposite of each term.

$$2-18x^{2}-9x^{2}+6x-1=1$$

Combine -18x^{2} and -9x^{2} to get -27x^{2}.

$$2-27x^{2}+6x-1=1$$

Subtract 1 from 2 to get 1.

$$1-27x^{2}+6x=1$$

Subtract 1 from both sides.

$$-27x^{2}+6x=1-1$$

Subtract 1 from 1 to get 0.

$$-27x^{2}+6x=0$$

Divide both sides by -27.

$$\frac{-27x^{2}+6x}{-27}=\frac{0}{-27}$$

Dividing by -27 undoes the multiplication by -27.

$$x^{2}+\frac{6}{-27}x=\frac{0}{-27}$$

Reduce the fraction \frac{6}{-27} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{2}{9}x=\frac{0}{-27}$$

Divide 0 by -27.

$$x^{2}-\frac{2}{9}x=0$$

Divide -\frac{2}{9}, the coefficient of the x term, by 2 to get -\frac{1}{9}. Then add the square of -\frac{1}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{9}x+\left(-\frac{1}{9}\right)^{2}=\left(-\frac{1}{9}\right)^{2}$$

Square -\frac{1}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{9}x+\frac{1}{81}=\frac{1}{81}$$

Factor x^{2}-\frac{2}{9}x+\frac{1}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{9}\right)^{2}=\frac{1}{81}$$

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{1}{9}=\frac{1}{9} x-\frac{1}{9}=-\frac{1}{9}$$

Add \frac{1}{9} to both sides of the equation.

$$x=\frac{2}{9} x=0$$