Solve 2017^2+2a^2+2016^2-8066a=1

$$2017 ^ { 2 } + 2 a ^ { 2 } + 2016 ^ { 2 } - 8066 a = 1$$

Solve for a

$$a=2016$$

$$a=2017$$

Steps Using the Quadratic Formula

Calculate 2017 to the power of 2 and get 4068289.

$$4068289+2a^{2}+2016^{2}-8066a=1$$

Calculate 2016 to the power of 2 and get 4064256.

$$4068289+2a^{2}[phone]a=1$$

Add 4068289 and 4064256 to get 8132545.

$$8132545+2a^{2}-8066a=1$$

Subtract 1 from both sides.

$$8132545+2a^{2}-8066a-1=0$$

Subtract 1 from 8132545 to get 8132544.

$$8132544+2a^{2}-8066a=0$$

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.

$$2a^{2}-8066a+8132544=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -8066 for b, and 8132544 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-8066\right)±\sqrt{\left(-8066\right)^{2}-4\times 2\times 8132544}}{2\times 2}$$

Square -8066.

$$a=\frac{-\left(-8066\right)±\sqrt{65060356-4\times 2\times 8132544}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-\left(-8066\right)±\sqrt{65060356-8\times 8132544}}{2\times 2}$$

Multiply -8 times 8132544.

$$a=\frac{-\left(-8066\right)±\sqrt{65060356-65060352}}{2\times 2}$$

Add 65060356 to -65060352.

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

Take the square root of 4.

$$a=\frac{-\left(-8066\right)±2}{2\times 2}$$

The opposite of -8066 is 8066.

$$a=\frac{8066±2}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{8066±2}{4}$$

Now solve the equation a=\frac{8066±2}{4} when ± is plus. Add 8066 to 2.

$$a=\frac{8068}{4}$$

Divide 8068 by 4.

$$a=2017$$

Now solve the equation a=\frac{8066±2}{4} when ± is minus. Subtract 2 from 8066.

$$a=\frac{8064}{4}$$

Divide 8064 by 4.

$$a=2016$$

The equation is now solved.

$$a=2017 a=2016$$

Steps for Completing the Square

Calculate 2017 to the power of 2 and get 4068289.

$$4068289+2a^{2}+2016^{2}-8066a=1$$

Calculate 2016 to the power of 2 and get 4064256.

$$4068289+2a^{2}[phone]a=1$$

Add 4068289 and 4064256 to get 8132545.

$$8132545+2a^{2}-8066a=1$$

Subtract 8132545 from both sides.

$$2a^{2}-8066a=1-8132545$$

Subtract 8132545 from 1 to get -8132544.

$$2a^{2}-8066a=-8132544$$

Divide both sides by 2.

$$\frac{2a^{2}-8066a}{2}=-\frac{8132544}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a^{2}+\left(-\frac{8066}{2}\right)a=-\frac{8132544}{2}$$

Divide -8066 by 2.

$$a^{2}-4033a=-\frac{8132544}{2}$$

Divide -8132544 by 2.

$$a^{2}-4033a=-4066272$$

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.

$$a^{2}-4033a+\left(-\frac{4033}{2}\right)^{2}=-4066272+\left(-\frac{4033}{2}\right)^{2}$$

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

$$a^{2}-4033a+\frac{16265089}{4}=-4066272+\frac{16265089}{4}$$

Add -4066272 to \frac{16265089}{4}.

$$a^{2}-4033a+\frac{16265089}{4}=\frac{1}{4}$$

Factor a^{2}-4033a+\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}.

$$\left(a-\frac{4033}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{4033}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$a-\frac{4033}{2}=\frac{1}{2} a-\frac{4033}{2}=-\frac{1}{2}$$

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

$$a=2017 a=2016$$