Solve (x-3)^2-4(2-x)(x+2)+(x-1)^3=x(x+4)(4-x)+2x^3-9x-46

$$( x - 3 ) ^ { 2 } - 4 ( 2 - x ) ( x + 2 ) + ( x - 1 ) ^ { 3 } = x ( x + 4 ) ( 4 - x ) + 2 x ^ { 3 } - 9 x - 46$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{51}i}{2}\approx 2.5+3.570714214i$$

$$x=\frac{-\sqrt{51}i+5}{2}\approx 2.5-3.570714214i$$

Steps Using the Quadratic Formula

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+\left(x-1\right)^{3}=x\left(x+4\right)\left(4-x\right)+2x^{3}-9x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=x\left(x+4\right)\left(4-x\right)+2x^{3}-9x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=\left(x^{2}+4x\right)\left(4-x\right)+2x^{3}-9x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=-x^{3}+16x+2x^{3}-9x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=x^{3}+16x-9x-46$$

Combine 16x and -9x to get 7x.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=x^{3}+7x-46$$

Subtract x^{3} from both sides.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1-x^{3}=7x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}+3x-1=7x-46$$

Subtract 7x from both sides.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}+3x-1-7x=-46$$

Combine 3x and -7x to get -4x.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}-4x-1=-46$$

Add 46 to both sides.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}-4x-1+46=0$$

Add -1 and 46 to get 45.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}-4x+45=0$$

Use the distributive property to multiply -4 by 2-x.

$$x^{2}-6x+9+\left(-8+4x\right)\left(x+2\right)-3x^{2}-4x+45=0$$

Use the distributive property to multiply -8+4x by x+2 and combine like terms.

$$x^{2}-6x+9-16+4x^{2}-3x^{2}-4x+45=0$$

Subtract 16 from 9 to get -7.

$$x^{2}-6x-7+4x^{2}-3x^{2}-4x+45=0$$

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

$$5x^{2}-6x-7-3x^{2}-4x+45=0$$

Combine 5x^{2} and -3x^{2} to get 2x^{2}.

$$2x^{2}-6x-7-4x+45=0$$

Combine -6x and -4x to get -10x.

$$2x^{2}-10x-7+45=0$$

Add -7 and 45 to get 38.

$$2x^{2}-10x+38=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -10 for b, and 38 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 2\times 38}}{2\times 2}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 2\times 38}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-10\right)±\sqrt{100-8\times 38}}{2\times 2}$$

Multiply -8 times 38.

$$x=\frac{-\left(-10\right)±\sqrt{100-304}}{2\times 2}$$

Add 100 to -304.

$$x=\frac{-\left(-10\right)±\sqrt{-204}}{2\times 2}$$

Take the square root of -204.

$$x=\frac{-\left(-10\right)±2\sqrt{51}i}{2\times 2}$$

The opposite of -10 is 10.

$$x=\frac{10±2\sqrt{51}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{10±2\sqrt{51}i}{4}$$

Now solve the equation x=\frac{10±2\sqrt{51}i}{4} when ± is plus. Add 10 to 2i\sqrt{51}.

$$x=\frac{10+2\sqrt{51}i}{4}$$

Divide 10+2i\sqrt{51} by 4.

$$x=\frac{5+\sqrt{51}i}{2}$$

Now solve the equation x=\frac{10±2\sqrt{51}i}{4} when ± is minus. Subtract 2i\sqrt{51} from 10.

$$x=\frac{-2\sqrt{51}i+10}{4}$$

Divide 10-2i\sqrt{51} by 4.

$$x=\frac{-\sqrt{51}i+5}{2}$$

The equation is now solved.

$$x=\frac{5+\sqrt{51}i}{2} x=\frac{-\sqrt{51}i+5}{2}$$

Steps for Completing the Square

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+\left(x-1\right)^{3}=x\left(x+4\right)\left(4-x\right)+2x^{3}-9x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=x\left(x+4\right)\left(4-x\right)+2x^{3}-9x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=\left(x^{2}+4x\right)\left(4-x\right)+2x^{3}-9x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=-x^{3}+16x+2x^{3}-9x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=x^{3}+16x-9x-46$$

Combine 16x and -9x to get 7x.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1=x^{3}+7x-46$$

Subtract x^{3} from both sides.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)+x^{3}-3x^{2}+3x-1-x^{3}=7x-46$$

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

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}+3x-1=7x-46$$

Subtract 7x from both sides.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}+3x-1-7x=-46$$

Combine 3x and -7x to get -4x.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}-4x-1=-46$$

Add 1 to both sides.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}-4x=-46+1$$

Add -46 and 1 to get -45.

$$x^{2}-6x+9-4\left(2-x\right)\left(x+2\right)-3x^{2}-4x=-45$$

Use the distributive property to multiply -4 by 2-x.

$$x^{2}-6x+9+\left(-8+4x\right)\left(x+2\right)-3x^{2}-4x=-45$$

Use the distributive property to multiply -8+4x by x+2 and combine like terms.

$$x^{2}-6x+9-16+4x^{2}-3x^{2}-4x=-45$$

Subtract 16 from 9 to get -7.

$$x^{2}-6x-7+4x^{2}-3x^{2}-4x=-45$$

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

$$5x^{2}-6x-7-3x^{2}-4x=-45$$

Combine 5x^{2} and -3x^{2} to get 2x^{2}.

$$2x^{2}-6x-7-4x=-45$$

Combine -6x and -4x to get -10x.

$$2x^{2}-10x-7=-45$$

Add 7 to both sides.

$$2x^{2}-10x=-45+7$$

Add -45 and 7 to get -38.

$$2x^{2}-10x=-38$$

Divide both sides by 2.

$$\frac{2x^{2}-10x}{2}=-\frac{38}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{10}{2}\right)x=-\frac{38}{2}$$

Divide -10 by 2.

$$x^{2}-5x=-\frac{38}{2}$$

Divide -38 by 2.

$$x^{2}-5x=-19$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-19+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-19+\frac{25}{4}$$

Add -19 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=-\frac{51}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{51}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{51}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{51}i}{2} x-\frac{5}{2}=-\frac{\sqrt{51}i}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{5+\sqrt{51}i}{2} x=\frac{-\sqrt{51}i+5}{2}$$