Solve x-2+x+x^2+left(x+1right)^2+2^2x+2^9=60

$$x-2+x+ { x }^{ 2 } + { \left(x+1 \right) }^{ 2 } + { 2 }^{ 2 } x+ { 2 }^{ 9 } =60$$

Solve for x (complex solution)

$$x=\frac{\sqrt{886}i}{2}-2\approx -2+14.882876066i$$

$$x=-\frac{\sqrt{886}i}{2}-2\approx -2-14.882876066i$$

Steps Using the Quadratic Formula

Combine x and x to get 2x.

$$2x-2+x^{2}+\left(x+1\right)^{2}+2^{2}x+2^{9}=60$$

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

$$2x-2+x^{2}+x^{2}+2x+1+2^{2}x+2^{9}=60$$

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

$$2x-2+2x^{2}+2x+1+2^{2}x+2^{9}=60$$

Combine 2x and 2x to get 4x.

$$4x-2+2x^{2}+1+2^{2}x+2^{9}=60$$

Add -2 and 1 to get -1.

$$4x-1+2x^{2}+2^{2}x+2^{9}=60$$

Calculate 2 to the power of 2 and get 4.

$$4x-1+2x^{2}+4x+2^{9}=60$$

Combine 4x and 4x to get 8x.

$$8x-1+2x^{2}+2^{9}=60$$

Calculate 2 to the power of 9 and get 512.

$$8x-1+2x^{2}+512=60$$

Add -1 and 512 to get 511.

$$8x+511+2x^{2}=60$$

Subtract 60 from both sides.

$$8x+511+2x^{2}-60=0$$

Subtract 60 from 511 to get 451.

$$8x+451+2x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$2x^{2}+8x+451=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and 451 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\times 451}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\times 451}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\times 451}}{2\times 2}$$

Multiply -8 times 451.

$$x=\frac{-8±\sqrt{64-3608}}{2\times 2}$$

Add 64 to -3608.

$$x=\frac{-8±\sqrt{-3544}}{2\times 2}$$

Take the square root of -3544.

$$x=\frac{-8±2\sqrt{886}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±2\sqrt{886}i}{4}$$

Now solve the equation x=\frac{-8±2\sqrt{886}i}{4} when ± is plus. Add -8 to 2i\sqrt{886}.

$$x=\frac{-8+2\sqrt{886}i}{4}$$

Divide -8+2i\sqrt{886} by 4.

$$x=\frac{\sqrt{886}i}{2}-2$$

Now solve the equation x=\frac{-8±2\sqrt{886}i}{4} when ± is minus. Subtract 2i\sqrt{886} from -8.

$$x=\frac{-2\sqrt{886}i-8}{4}$$

Divide -8-2i\sqrt{886} by 4.

$$x=-\frac{\sqrt{886}i}{2}-2$$

The equation is now solved.

$$x=\frac{\sqrt{886}i}{2}-2 x=-\frac{\sqrt{886}i}{2}-2$$

Steps for Completing the Square

Combine x and x to get 2x.

$$2x-2+x^{2}+\left(x+1\right)^{2}+2^{2}x+2^{9}=60$$

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

$$2x-2+x^{2}+x^{2}+2x+1+2^{2}x+2^{9}=60$$

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

$$2x-2+2x^{2}+2x+1+2^{2}x+2^{9}=60$$

Combine 2x and 2x to get 4x.

$$4x-2+2x^{2}+1+2^{2}x+2^{9}=60$$

Add -2 and 1 to get -1.

$$4x-1+2x^{2}+2^{2}x+2^{9}=60$$

Calculate 2 to the power of 2 and get 4.

$$4x-1+2x^{2}+4x+2^{9}=60$$

Combine 4x and 4x to get 8x.

$$8x-1+2x^{2}+2^{9}=60$$

Calculate 2 to the power of 9 and get 512.

$$8x-1+2x^{2}+512=60$$

Add -1 and 512 to get 511.

$$8x+511+2x^{2}=60$$

Subtract 511 from both sides.

$$8x+2x^{2}=60-511$$

Subtract 511 from 60 to get -451.

$$8x+2x^{2}=-451$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$2x^{2}+8x=-451$$

Divide both sides by 2.

$$\frac{2x^{2}+8x}{2}=-\frac{451}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{8}{2}x=-\frac{451}{2}$$

Divide 8 by 2.

$$x^{2}+4x=-\frac{451}{2}$$

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

$$x^{2}+4x+2^{2}=-\frac{451}{2}+2^{2}$$

Square 2.

$$x^{2}+4x+4=-\frac{451}{2}+4$$

Add -\frac{451}{2} to 4.

$$x^{2}+4x+4=-\frac{443}{2}$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=-\frac{443}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{-\frac{443}{2}}$$

Simplify.

$$x+2=\frac{\sqrt{886}i}{2} x+2=-\frac{\sqrt{886}i}{2}$$

Subtract 2 from both sides of the equation.

$$x=\frac{\sqrt{886}i}{2}-2 x=-\frac{\sqrt{886}i}{2}-2$$