Solve (x-1)^3-x(x-1)^2=5x(2-x)-11(x+2)

$$( x - 1 ) ^ { 3 } - x ( x - 1 ) ^ { 2 } = 5 x ( 2 - x ) - 11 ( x + 2 )$$

Solve for x (complex solution)

$$x=\frac{-3+\sqrt{327}i}{8}\approx -0.375+2.260392665i$$

$$x=\frac{-\sqrt{327}i-3}{8}\approx -0.375-2.260392665i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$x^{3}-3x^{2}+3x-1-x\left(x-1\right)^{2}=5x\left(2-x\right)-11\left(x+2\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{3}-3x^{2}+3x-1-x\left(x^{2}-2x+1\right)=5x\left(2-x\right)-11\left(x+2\right)$$

Use the distributive property to multiply x by x^{2}-2x+1.

$$x^{3}-3x^{2}+3x-1-\left(x^{3}-2x^{2}+x\right)=5x\left(2-x\right)-11\left(x+2\right)$$

To find the opposite of x^{3}-2x^{2}+x, find the opposite of each term.

$$x^{3}-3x^{2}+3x-1-x^{3}+2x^{2}-x=5x\left(2-x\right)-11\left(x+2\right)$$

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

$$-3x^{2}+3x-1+2x^{2}-x=5x\left(2-x\right)-11\left(x+2\right)$$

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

$$-x^{2}+3x-1-x=5x\left(2-x\right)-11\left(x+2\right)$$

Combine 3x and -x to get 2x.

$$-x^{2}+2x-1=5x\left(2-x\right)-11\left(x+2\right)$$

Use the distributive property to multiply 5x by 2-x.

$$-x^{2}+2x-1=10x-5x^{2}-11\left(x+2\right)$$

Use the distributive property to multiply -11 by x+2.

$$-x^{2}+2x-1=10x-5x^{2}-11x-22$$

Combine 10x and -11x to get -x.

$$-x^{2}+2x-1=-x-5x^{2}-22$$

Add x to both sides.

$$-x^{2}+2x-1+x=-5x^{2}-22$$

Combine 2x and x to get 3x.

$$-x^{2}+3x-1=-5x^{2}-22$$

Add 5x^{2} to both sides.

$$-x^{2}+3x-1+5x^{2}=-22$$

Combine -x^{2} and 5x^{2} to get 4x^{2}.

$$4x^{2}+3x-1=-22$$

Add 22 to both sides.

$$4x^{2}+3x-1+22=0$$

Add -1 and 22 to get 21.

$$4x^{2}+3x+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 3 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 4\times 21}}{2\times 4}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 4\times 21}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-3±\sqrt{9-16\times 21}}{2\times 4}$$

Multiply -16 times 21.

$$x=\frac{-3±\sqrt{9-336}}{2\times 4}$$

Add 9 to -336.

$$x=\frac{-3±\sqrt{-327}}{2\times 4}$$

Take the square root of -327.

$$x=\frac{-3±\sqrt{327}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-3±\sqrt{327}i}{8}$$

Now solve the equation x=\frac{-3±\sqrt{327}i}{8} when ± is plus. Add -3 to i\sqrt{327}.

$$x=\frac{-3+\sqrt{327}i}{8}$$

Now solve the equation x=\frac{-3±\sqrt{327}i}{8} when ± is minus. Subtract i\sqrt{327} from -3.

$$x=\frac{-\sqrt{327}i-3}{8}$$

The equation is now solved.

$$x=\frac{-3+\sqrt{327}i}{8} x=\frac{-\sqrt{327}i-3}{8}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$x^{3}-3x^{2}+3x-1-x\left(x-1\right)^{2}=5x\left(2-x\right)-11\left(x+2\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{3}-3x^{2}+3x-1-x\left(x^{2}-2x+1\right)=5x\left(2-x\right)-11\left(x+2\right)$$

Use the distributive property to multiply x by x^{2}-2x+1.

$$x^{3}-3x^{2}+3x-1-\left(x^{3}-2x^{2}+x\right)=5x\left(2-x\right)-11\left(x+2\right)$$

To find the opposite of x^{3}-2x^{2}+x, find the opposite of each term.

$$x^{3}-3x^{2}+3x-1-x^{3}+2x^{2}-x=5x\left(2-x\right)-11\left(x+2\right)$$

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

$$-3x^{2}+3x-1+2x^{2}-x=5x\left(2-x\right)-11\left(x+2\right)$$

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

$$-x^{2}+3x-1-x=5x\left(2-x\right)-11\left(x+2\right)$$

Combine 3x and -x to get 2x.

$$-x^{2}+2x-1=5x\left(2-x\right)-11\left(x+2\right)$$

Use the distributive property to multiply 5x by 2-x.

$$-x^{2}+2x-1=10x-5x^{2}-11\left(x+2\right)$$

Use the distributive property to multiply -11 by x+2.

$$-x^{2}+2x-1=10x-5x^{2}-11x-22$$

Combine 10x and -11x to get -x.

$$-x^{2}+2x-1=-x-5x^{2}-22$$

Add x to both sides.

$$-x^{2}+2x-1+x=-5x^{2}-22$$

Combine 2x and x to get 3x.

$$-x^{2}+3x-1=-5x^{2}-22$$

Add 5x^{2} to both sides.

$$-x^{2}+3x-1+5x^{2}=-22$$

Combine -x^{2} and 5x^{2} to get 4x^{2}.

$$4x^{2}+3x-1=-22$$

Add 1 to both sides.

$$4x^{2}+3x=-22+1$$

Add -22 and 1 to get -21.

$$4x^{2}+3x=-21$$

Divide both sides by 4.

$$\frac{4x^{2}+3x}{4}=-\frac{21}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{3}{4}x=-\frac{21}{4}$$

Divide \frac{3}{4}, the coefficient of the x term, by 2 to get \frac{3}{8}. Then add the square of \frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=-\frac{21}{4}+\left(\frac{3}{8}\right)^{2}$$

Square \frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{21}{4}+\frac{9}{64}$$

Add -\frac{21}{4} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=-\frac{327}{64}$$

Factor x^{2}+\frac{3}{4}x+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{8}\right)^{2}=-\frac{327}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{-\frac{327}{64}}$$

Simplify.

$$x+\frac{3}{8}=\frac{\sqrt{327}i}{8} x+\frac{3}{8}=-\frac{\sqrt{327}i}{8}$$

Subtract \frac{3}{8} from both sides of the equation.

$$x=\frac{-3+\sqrt{327}i}{8} x=\frac{-\sqrt{327}i-3}{8}$$