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

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

Solve for x

$$x=-1$$

$$x=3$$

Steps Using the Quadratic Formula

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

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

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

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

Subtract 2x^{2} from both sides.

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

Combine 3x^{2} and -2x^{2} to get x^{2}.

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

Subtract 5x from both sides.

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

Combine 3x and -5x to get -2x.

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

Subtract 3 from both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -2.

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

Multiply -4 times -3.

$$x=\frac{-\left(-2\right)±\sqrt{4+12}}{2}$$

Add 4 to 12.

$$x=\frac{-\left(-2\right)±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-\left(-2\right)±4}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±4}{2}$$

Now solve the equation x=\frac{2±4}{2} when ± is plus. Add 2 to 4.

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

Divide 6 by 2.

$$x=3$$

Now solve the equation x=\frac{2±4}{2} when ± is minus. Subtract 4 from 2.

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

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=3 x=-1$$

Steps for Completing the Square

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

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

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

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

Subtract 2x^{2} from both sides.

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

Combine 3x^{2} and -2x^{2} to get x^{2}.

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

Subtract 5x from both sides.

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

Combine 3x and -5x to get -2x.

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

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

$$x^{2}-2x+1=3+1$$

Add 3 to 1.

$$x^{2}-2x+1=4$$

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

$$\left(x-1\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-1=2 x-1=-2$$

Add 1 to both sides of the equation.

$$x=3 x=-1$$