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

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

Solve for x

$$x=\frac{\sqrt{14}-3}{5}\approx 0.148331477$$

$$x=\frac{-\sqrt{14}-3}{5}\approx -1.348331477$$

Steps Using the Quadratic Formula

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

$$6x^{2}+7x-3=\left(x+2\right)\left(x-1\right)$$

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

$$6x^{2}+7x-3=x^{2}+x-2$$

Subtract x^{2} from both sides.

$$6x^{2}+7x-3-x^{2}=x-2$$

Combine 6x^{2} and -x^{2} to get 5x^{2}.

$$5x^{2}+7x-3=x-2$$

Subtract x from both sides.

$$5x^{2}+7x-3-x=-2$$

Combine 7x and -x to get 6x.

$$5x^{2}+6x-3=-2$$

Add 2 to both sides.

$$5x^{2}+6x-3+2=0$$

Add -3 and 2 to get -1.

$$5x^{2}+6x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 6 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 5\left(-1\right)}}{2\times 5}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 5\left(-1\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-6±\sqrt{36-20\left(-1\right)}}{2\times 5}$$

Multiply -20 times -1.

$$x=\frac{-6±\sqrt{36+20}}{2\times 5}$$

Add 36 to 20.

$$x=\frac{-6±\sqrt{56}}{2\times 5}$$

Take the square root of 56.

$$x=\frac{-6±2\sqrt{14}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-6±2\sqrt{14}}{10}$$

Now solve the equation x=\frac{-6±2\sqrt{14}}{10} when ± is plus. Add -6 to 2\sqrt{14}.

$$x=\frac{2\sqrt{14}-6}{10}$$

Divide -6+2\sqrt{14} by 10.

$$x=\frac{\sqrt{14}-3}{5}$$

Now solve the equation x=\frac{-6±2\sqrt{14}}{10} when ± is minus. Subtract 2\sqrt{14} from -6.

$$x=\frac{-2\sqrt{14}-6}{10}$$

Divide -6-2\sqrt{14} by 10.

$$x=\frac{-\sqrt{14}-3}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{14}-3}{5} x=\frac{-\sqrt{14}-3}{5}$$

Steps for Completing the Square

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

$$6x^{2}+7x-3=\left(x+2\right)\left(x-1\right)$$

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

$$6x^{2}+7x-3=x^{2}+x-2$$

Subtract x^{2} from both sides.

$$6x^{2}+7x-3-x^{2}=x-2$$

Combine 6x^{2} and -x^{2} to get 5x^{2}.

$$5x^{2}+7x-3=x-2$$

Subtract x from both sides.

$$5x^{2}+7x-3-x=-2$$

Combine 7x and -x to get 6x.

$$5x^{2}+6x-3=-2$$

Add 3 to both sides.

$$5x^{2}+6x=-2+3$$

Add -2 and 3 to get 1.

$$5x^{2}+6x=1$$

Divide both sides by 5.

$$\frac{5x^{2}+6x}{5}=\frac{1}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{6}{5}x=\frac{1}{5}$$

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

$$x^{2}+\frac{6}{5}x+\left(\frac{3}{5}\right)^{2}=\frac{1}{5}+\left(\frac{3}{5}\right)^{2}$$

Square \frac{3}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{1}{5}+\frac{9}{25}$$

Add \frac{1}{5} to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{14}{25}$$

Factor x^{2}+\frac{6}{5}x+\frac{9}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{5}\right)^{2}=\frac{14}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{5}\right)^{2}}=\sqrt{\frac{14}{25}}$$

Simplify.

$$x+\frac{3}{5}=\frac{\sqrt{14}}{5} x+\frac{3}{5}=-\frac{\sqrt{14}}{5}$$

Subtract \frac{3}{5} from both sides of the equation.

$$x=\frac{\sqrt{14}-3}{5} x=\frac{-\sqrt{14}-3}{5}$$