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4068288-4034x+x^{2}=2017
Use the distributive property to multiply 2016-x by 2018-x and combine like terms.
4068288-4034x+x^{2}-2017=0
Subtract 2017 from both sides.
4066271-4034x+x^{2}=0
Subtract 2017 from 4068288 to get 4066271.
x^{2}-4034x+4066271=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.
x=\frac{-\left(-4034\right)±\sqrt{\left(-4034\right)^{2}-4\times 4066271}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4034 for b, and 4066271 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-4034\right)±\sqrt{16273156-4\times 4066271}}{2}
Square -4034.
x=\frac{-\left(-4034\right)±\sqrt{16273156-16265084}}{2}
Multiply -4 times 4066271.
x=\frac{-\left(-4034\right)±\sqrt{8072}}{2}
Add 16273156 to -16265084.
x=\frac{-\left(-4034\right)±2\sqrt{2018}}{2}
Take the square root of 8072.
x=\frac{4034±2\sqrt{2018}}{2}
The opposite of -4034 is 4034.
x=\frac{2\sqrt{2018}+4034}{2}
Now solve the equation x=\frac{4034±2\sqrt{2018}}{2} when ± is plus. Add 4034 to 2\sqrt{2018}.
x=\sqrt{2018}+2017
Divide 4034+2\sqrt{2018} by 2.
x=\frac{4034-2\sqrt{2018}}{2}
Now solve the equation x=\frac{4034±2\sqrt{2018}}{2} when ± is minus. Subtract 2\sqrt{2018} from 4034.
x=2017-\sqrt{2018}
Divide 4034-2\sqrt{2018} by 2.
x=\sqrt{2018}+2017 x=2017-\sqrt{2018}
The equation is now solved.
4068288-4034x+x^{2}=2017
Use the distributive property to multiply 2016-x by 2018-x and combine like terms.
-4034x+x^{2}=2017-4068288
Subtract 4068288 from both sides.
-4034x+x^{2}=-4066271
Subtract 4068288 from 2017 to get -4066271.
x^{2}-4034x=-4066271
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.
x^{2}-4034x+\left(-2017\right)^{2}=-4066271+\left(-2017\right)^{2}
Divide -4034, the coefficient of the x term, by 2 to get -2017. Then add the square of -2017 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-4034x+4068289=-4066271+4068289
Square -2017.
x^{2}-4034x+4068289=2018
Add -4066271 to 4068289.
\left(x-2017\right)^{2}=2018
Factor x^{2}-4034x+4068289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-2017\right)^{2}}=\sqrt{2018}
Take the square root of both sides of the equation.
x-2017=\sqrt{2018} x-2017=-\sqrt{2018}
Simplify.
x=\sqrt{2018}+2017 x=2017-\sqrt{2018}
Add 2017 to both sides of the equation.