Solve xleft(x+1right)/3-1=(x-1)left(x+2right)/4

$$\frac{ x \left( x+1 \right) }{ 3 } -1= \frac{ (x-1) \left( x+2 \right) }{ 4 }$$

Solve for x

$$x=-3$$

$$x=2$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4x\left(x+1\right)-12=3\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 4x by x+1.

$$4x^{2}+4x-12=3\left(x-1\right)\left(x+2\right)$$

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

$$4x^{2}+4x-12=\left(3x-3\right)\left(x+2\right)$$

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

$$4x^{2}+4x-12=3x^{2}+3x-6$$

Subtract 3x^{2} from both sides.

$$4x^{2}+4x-12-3x^{2}=3x-6$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}+4x-12=3x-6$$

Subtract 3x from both sides.

$$x^{2}+4x-12-3x=-6$$

Combine 4x and -3x to get x.

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

Add 6 to both sides.

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

Add -12 and 6 to get -6.

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

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

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

Square 1.

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

Multiply -4 times -6.

$$x=\frac{-1±\sqrt{1+24}}{2}$$

Add 1 to 24.

$$x=\frac{-1±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-1±5}{2}$$

Now solve the equation x=\frac{-1±5}{2} when ± is plus. Add -1 to 5.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-1±5}{2} when ± is minus. Subtract 5 from -1.

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

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=2 x=-3$$

Steps for Completing the Square

Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4x\left(x+1\right)-12=3\left(x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 4x by x+1.

$$4x^{2}+4x-12=3\left(x-1\right)\left(x+2\right)$$

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

$$4x^{2}+4x-12=\left(3x-3\right)\left(x+2\right)$$

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

$$4x^{2}+4x-12=3x^{2}+3x-6$$

Subtract 3x^{2} from both sides.

$$4x^{2}+4x-12-3x^{2}=3x-6$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}+4x-12=3x-6$$

Subtract 3x from both sides.

$$x^{2}+4x-12-3x=-6$$

Combine 4x and -3x to get x.

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

Add 12 to both sides.

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

Add -6 and 12 to get 6.

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

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=6+\frac{1}{4}$$

Add 6 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{25}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{5}{2} x+\frac{1}{2}=-\frac{5}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=2 x=-3$$