Solve x(x-1)(x-3)=(x+1)(x+2)(x+4)+3

$$x ( x - 1 ) ( x - 3 ) = ( x + 1 ) ( x + 2 ) ( x + 4 ) + 3$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{3}i}{2}\approx -0.5+0.866025404i$$

$$x=\frac{-\sqrt{3}i-1}{2}\approx -0.5-0.866025404i$$

Steps Using the Quadratic Formula

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

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

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

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

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

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

Use the distributive property to multiply x^{2}+3x+2 by x+4 and combine like terms.

$$x^{3}-4x^{2}+3x=x^{3}+7x^{2}+14x+8+3$$

Add 8 and 3 to get 11.

$$x^{3}-4x^{2}+3x=x^{3}+7x^{2}+14x+11$$

Subtract x^{3} from both sides.

$$x^{3}-4x^{2}+3x-x^{3}=7x^{2}+14x+11$$

Combine x^{3} and -x^{3} to get 0.

$$-4x^{2}+3x=7x^{2}+14x+11$$

Subtract 7x^{2} from both sides.

$$-4x^{2}+3x-7x^{2}=14x+11$$

Combine -4x^{2} and -7x^{2} to get -11x^{2}.

$$-11x^{2}+3x=14x+11$$

Subtract 14x from both sides.

$$-11x^{2}+3x-14x=11$$

Combine 3x and -14x to get -11x.

$$-11x^{2}-11x=11$$

Subtract 11 from both sides.

$$-11x^{2}-11x-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -11 for a, -11 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-11\right)\left(-11\right)}}{2\left(-11\right)}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-11\right)\left(-11\right)}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$x=\frac{-\left(-11\right)±\sqrt{121+44\left(-11\right)}}{2\left(-11\right)}$$

Multiply 44 times -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-484}}{2\left(-11\right)}$$

Add 121 to -484.

$$x=\frac{-\left(-11\right)±\sqrt{-363}}{2\left(-11\right)}$$

Take the square root of -363.

$$x=\frac{-\left(-11\right)±11\sqrt{3}i}{2\left(-11\right)}$$

The opposite of -11 is 11.

$$x=\frac{11±11\sqrt{3}i}{2\left(-11\right)}$$

Multiply 2 times -11.

$$x=\frac{11±11\sqrt{3}i}{-22}$$

Now solve the equation x=\frac{11±11\sqrt{3}i}{-22} when ± is plus. Add 11 to 11i\sqrt{3}.

$$x=\frac{11+11\sqrt{3}i}{-22}$$

Divide 11+11i\sqrt{3} by -22.

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

Now solve the equation x=\frac{11±11\sqrt{3}i}{-22} when ± is minus. Subtract 11i\sqrt{3} from 11.

$$x=\frac{-11\sqrt{3}i+11}{-22}$$

Divide 11-11i\sqrt{3} by -22.

$$x=\frac{-1+\sqrt{3}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{3}i-1}{2} x=\frac{-1+\sqrt{3}i}{2}$$

Steps for Completing the Square

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

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

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

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

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

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

Use the distributive property to multiply x^{2}+3x+2 by x+4 and combine like terms.

$$x^{3}-4x^{2}+3x=x^{3}+7x^{2}+14x+8+3$$

Add 8 and 3 to get 11.

$$x^{3}-4x^{2}+3x=x^{3}+7x^{2}+14x+11$$

Subtract x^{3} from both sides.

$$x^{3}-4x^{2}+3x-x^{3}=7x^{2}+14x+11$$

Combine x^{3} and -x^{3} to get 0.

$$-4x^{2}+3x=7x^{2}+14x+11$$

Subtract 7x^{2} from both sides.

$$-4x^{2}+3x-7x^{2}=14x+11$$

Combine -4x^{2} and -7x^{2} to get -11x^{2}.

$$-11x^{2}+3x=14x+11$$

Subtract 14x from both sides.

$$-11x^{2}+3x-14x=11$$

Combine 3x and -14x to get -11x.

$$-11x^{2}-11x=11$$

Divide both sides by -11.

$$\frac{-11x^{2}-11x}{-11}=\frac{11}{-11}$$

Dividing by -11 undoes the multiplication by -11.

$$x^{2}+\left(-\frac{11}{-11}\right)x=\frac{11}{-11}$$

Divide -11 by -11.

$$x^{2}+x=\frac{11}{-11}$$

Divide 11 by -11.

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

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.

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

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

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

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

$$x^{2}+x+\frac{1}{4}=-\frac{3}{4}$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{3}i}{2} x+\frac{1}{2}=-\frac{\sqrt{3}i}{2}$$

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

$$x=\frac{-1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i-1}{2}$$