Solve [(2018-x)+(2015-x)]^2+3289=2(2018-x)(2015-x)

$$[ ( 2018 - x ) + ( 2015 - x ) ] ^ { 2 } + 3289 = 2 ( 2018 - x ) ( 2015 - x )$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{6587}i+4033}{2}\approx 2016.5-40.58016757i$$

$$x=\frac{4033+\sqrt{6587}i}{2}\approx 2016.5+40.58016757i$$

Steps Using the Quadratic Formula

Add 2018 and 2015 to get 4033.

$$\left(4033-x-x\right)^{2}+3289=2\left(2018-x\right)\left(2015-x\right)$$

Combine -x and -x to get -2x.

$$\left(4033-2x\right)^{2}+3289=2\left(2018-x\right)\left(2015-x\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4033-2x\right)^{2}.

$$16265089-16132x+4x^{2}+3289=2\left(2018-x\right)\left(2015-x\right)$$

Add 16265089 and 3289 to get 16268378.

$$16268378-16132x+4x^{2}=2\left(2018-x\right)\left(2015-x\right)$$

Use the distributive property to multiply 2 by 2018-x.

$$16268378-16132x+4x^{2}=\left(4036-2x\right)\left(2015-x\right)$$

Use the distributive property to multiply 4036-2x by 2015-x and combine like terms.

$$16268378-16132x+4x^{2}=8132540-8066x+2x^{2}$$

Subtract 8132540 from both sides.

$$16268378-16132x+4x^{2}-8132540=-8066x+2x^{2}$$

Subtract 8132540 from 16268378 to get 8135838.

$$8135838-16132x+4x^{2}=-8066x+2x^{2}$$

Add 8066x to both sides.

$$8135838-16132x+4x^{2}+8066x=2x^{2}$$

Combine -16132x and 8066x to get -8066x.

$$8135838-8066x+4x^{2}=2x^{2}$$

Subtract 2x^{2} from both sides.

$$8135838-8066x+4x^{2}-2x^{2}=0$$

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

$$8135838-8066x+2x^{2}=0$$

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.

$$2x^{2}-8066x+8135838=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -8066 for b, and 8135838 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8066\right)±\sqrt{\left(-8066\right)^{2}-4\times 2\times 8135838}}{2\times 2}$$

Square -8066.

$$x=\frac{-\left(-8066\right)±\sqrt{65060356-4\times 2\times 8135838}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-8066\right)±\sqrt{65060356-8\times 8135838}}{2\times 2}$$

Multiply -8 times 8135838.

$$x=\frac{-\left(-8066\right)±\sqrt{65060356-65086704}}{2\times 2}$$

Add 65060356 to -65086704.

$$x=\frac{-\left(-8066\right)±\sqrt{-26348}}{2\times 2}$$

Take the square root of -26348.

$$x=\frac{-\left(-8066\right)±2\sqrt{6587}i}{2\times 2}$$

The opposite of -8066 is 8066.

$$x=\frac{8066±2\sqrt{6587}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{8066±2\sqrt{6587}i}{4}$$

Now solve the equation x=\frac{8066±2\sqrt{6587}i}{4} when ± is plus. Add 8066 to 2i\sqrt{6587}.

$$x=\frac{8066+2\sqrt{6587}i}{4}$$

Divide 8066+2i\sqrt{6587} by 4.

$$x=\frac{4033+\sqrt{6587}i}{2}$$

Now solve the equation x=\frac{8066±2\sqrt{6587}i}{4} when ± is minus. Subtract 2i\sqrt{6587} from 8066.

$$x=\frac{-2\sqrt{6587}i+8066}{4}$$

Divide 8066-2i\sqrt{6587} by 4.

$$x=\frac{-\sqrt{6587}i+4033}{2}$$

The equation is now solved.

$$x=\frac{4033+\sqrt{6587}i}{2} x=\frac{-\sqrt{6587}i+4033}{2}$$

Steps for Completing the Square

Add 2018 and 2015 to get 4033.

$$\left(4033-x-x\right)^{2}+3289=2\left(2018-x\right)\left(2015-x\right)$$

Combine -x and -x to get -2x.

$$\left(4033-2x\right)^{2}+3289=2\left(2018-x\right)\left(2015-x\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4033-2x\right)^{2}.

$$16265089-16132x+4x^{2}+3289=2\left(2018-x\right)\left(2015-x\right)$$

Add 16265089 and 3289 to get 16268378.

$$16268378-16132x+4x^{2}=2\left(2018-x\right)\left(2015-x\right)$$

Use the distributive property to multiply 2 by 2018-x.

$$16268378-16132x+4x^{2}=\left(4036-2x\right)\left(2015-x\right)$$

Use the distributive property to multiply 4036-2x by 2015-x and combine like terms.

$$16268378-16132x+4x^{2}=8132540-8066x+2x^{2}$$

Add 8066x to both sides.

$$16268378-16132x+4x^{2}+8066x=8132540+2x^{2}$$

Combine -16132x and 8066x to get -8066x.

$$16268378-8066x+4x^{2}=8132540+2x^{2}$$

Subtract 2x^{2} from both sides.

$$16268378-8066x+4x^{2}-2x^{2}=8132540$$

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

$$16268378-8066x+2x^{2}=8132540$$

Subtract 16268378 from both sides.

$$-8066x+2x^{2}=8132540-16268378$$

Subtract 16268378 from 8132540 to get -8135838.

$$-8066x+2x^{2}=-8135838$$

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.

$$2x^{2}-8066x=-8135838$$

Divide both sides by 2.

$$\frac{2x^{2}-8066x}{2}=-\frac{8135838}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{8066}{2}\right)x=-\frac{8135838}{2}$$

Divide -8066 by 2.

$$x^{2}-4033x=-\frac{8135838}{2}$$

Divide -8135838 by 2.

$$x^{2}-4033x=-4067919$$

Divide -4033, the coefficient of the x term, by 2 to get -\frac{4033}{2}. Then add the square of -\frac{4033}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4033x+\left(-\frac{4033}{2}\right)^{2}=-4067919+\left(-\frac{4033}{2}\right)^{2}$$

Square -\frac{4033}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-4033x+\frac{16265089}{4}=-4067919+\frac{16265089}{4}$$

Add -4067919 to \frac{16265089}{4}.

$$x^{2}-4033x+\frac{16265089}{4}=-\frac{6587}{4}$$

Factor x^{2}-4033x+\frac{16265089}{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}.

$$\left(x-\frac{4033}{2}\right)^{2}=-\frac{6587}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4033}{2}\right)^{2}}=\sqrt{-\frac{6587}{4}}$$

Simplify.

$$x-\frac{4033}{2}=\frac{\sqrt{6587}i}{2} x-\frac{4033}{2}=-\frac{\sqrt{6587}i}{2}$$

Add \frac{4033}{2} to both sides of the equation.

$$x=\frac{4033+\sqrt{6587}i}{2} x=\frac{-\sqrt{6587}i+4033}{2}$$