Solve 3x(x+1)=2(1-x)

$$3 x ( x + 1 ) = 2 ( 1 - x )$$

Solve for x

$$x=-2$$

$$x=\frac{1}{3}\approx 0.333333333$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by x+1.

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

Use the distributive property to multiply 2 by 1-x.

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

Subtract 2 from both sides.

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

Add 2x to both sides.

$$3x^{2}+3x-2+2x=0$$

Combine 3x and 2x to get 5x.

$$3x^{2}+5x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 5 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 3\left(-2\right)}}{2\times 3}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 3\left(-2\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-5±\sqrt{25-12\left(-2\right)}}{2\times 3}$$

Multiply -12 times -2.

$$x=\frac{-5±\sqrt{25+24}}{2\times 3}$$

Add 25 to 24.

$$x=\frac{-5±\sqrt{49}}{2\times 3}$$

Take the square root of 49.

$$x=\frac{-5±7}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-5±7}{6}$$

Now solve the equation x=\frac{-5±7}{6} when ± is plus. Add -5 to 7.

$$x=\frac{2}{6}$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{-5±7}{6} when ± is minus. Subtract 7 from -5.

$$x=-\frac{12}{6}$$

Divide -12 by 6.

$$x=-2$$

The equation is now solved.

$$x=\frac{1}{3} x=-2$$

Steps for Completing the Square

Use the distributive property to multiply 3x by x+1.

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

Use the distributive property to multiply 2 by 1-x.

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

Add 2x to both sides.

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

Combine 3x and 2x to get 5x.

$$3x^{2}+5x=2$$

Divide both sides by 3.

$$\frac{3x^{2}+5x}{3}=\frac{2}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{5}{3}x=\frac{2}{3}$$

Divide \frac{5}{3}, the coefficient of the x term, by 2 to get \frac{5}{6}. Then add the square of \frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=\frac{2}{3}+\left(\frac{5}{6}\right)^{2}$$

Square \frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{2}{3}+\frac{25}{36}$$

Add \frac{2}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=\frac{49}{36}$$

Factor x^{2}+\frac{5}{3}x+\frac{25}{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}.

$$\left(x+\frac{5}{6}\right)^{2}=\frac{49}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Simplify.

$$x+\frac{5}{6}=\frac{7}{6} x+\frac{5}{6}=-\frac{7}{6}$$

Subtract \frac{5}{6} from both sides of the equation.

$$x=\frac{1}{3} x=-2$$