Solve 2016-a+sqrt{a-2017}=a

$$2016 - a + \sqrt { a - 2017 } = a$$

Solve for a

$$a=\frac{8065+\sqrt{16143}i}{8}\approx 1008.125+15.881888269i$$

Solution Steps

Subtract 2016-a from both sides of the equation.

$$\sqrt{a-2017}=a-\left(2016-a\right)$$

To find the opposite of 2016-a, find the opposite of each term.

$$\sqrt{a-2017}=a-2016-\left(-a\right)$$

The opposite of -a is a.

$$\sqrt{a-2017}=a-2016+a$$

Combine a and a to get 2a.

$$\sqrt{a-2017}=2a-2016$$

Square both sides of the equation.

$$\left(\sqrt{a-2017}\right)^{2}=\left(2a-2016\right)^{2}$$

Calculate \sqrt{a-2017} to the power of 2 and get a-2017.

$$a-2017=\left(2a-2016\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2a-2016\right)^{2}.

$$a-2017=4a^{2}-8064a+4064256$$

Subtract 4a^{2} from both sides.

$$a-2017-4a^{2}=-8064a+4064256$$

Add 8064a to both sides.

$$a-2017-4a^{2}+8064a=4064256$$

Combine a and 8064a to get 8065a.

$$8065a-2017-4a^{2}=4064256$$

Subtract 4064256 from both sides.

$$8065a-2017-4a^{2}-4064256=0$$

Subtract 4064256 from -2017 to get -4066273.

$$8065a-4066273-4a^{2}=0$$

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.

$$-4a^{2}+8065a-4066273=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 8065 for b, and -4066273 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-8065±\sqrt{8065^{2}-4\left(-4\right)\left(-4066273\right)}}{2\left(-4\right)}$$

Square 8065.

$$a=\frac{-8065±\sqrt{65044225-4\left(-4\right)\left(-4066273\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$a=\frac{-8065±\sqrt{65044225+16\left(-4066273\right)}}{2\left(-4\right)}$$

Multiply 16 times -4066273.

$$a=\frac{-8065±\sqrt{65044225-65060368}}{2\left(-4\right)}$$

Add 65044225 to -65060368.

$$a=\frac{-8065±\sqrt{-16143}}{2\left(-4\right)}$$

Take the square root of -16143.

$$a=\frac{-8065±\sqrt{16143}i}{2\left(-4\right)}$$

Multiply 2 times -4.

$$a=\frac{-8065±\sqrt{16143}i}{-8}$$

Now solve the equation a=\frac{-8065±\sqrt{16143}i}{-8} when ± is plus. Add -8065 to i\sqrt{16143}.

$$a=\frac{-8065+\sqrt{16143}i}{-8}$$

Divide -8065+i\sqrt{16143} by -8.

$$a=\frac{-\sqrt{16143}i+8065}{8}$$

Now solve the equation a=\frac{-8065±\sqrt{16143}i}{-8} when ± is minus. Subtract i\sqrt{16143} from -8065.

$$a=\frac{-\sqrt{16143}i-8065}{-8}$$

Divide -8065-i\sqrt{16143} by -8.

$$a=\frac{8065+\sqrt{16143}i}{8}$$

The equation is now solved.

$$a=\frac{-\sqrt{16143}i+8065}{8} a=\frac{8065+\sqrt{16143}i}{8}$$

Substitute \frac{-\sqrt{16143}i+8065}{8} for a in the equation 2016-a+\sqrt{a-2017}=a.

$$2016-\frac{-\sqrt{16143}i+8065}{8}+\sqrt{\frac{-\sqrt{16143}i+8065}{8}-2017}=\frac{-\sqrt{16143}i+8065}{8}$$

Simplify. The value a=\frac{-\sqrt{16143}i+8065}{8} does not satisfy the equation.

$$\frac{8061}{8}+\frac{3}{8}i\times 16143^{\frac{1}{2}}=-\frac{1}{8}i\times 16143^{\frac{1}{2}}+\frac{8065}{8}$$

Substitute \frac{8065+\sqrt{16143}i}{8} for a in the equation 2016-a+\sqrt{a-2017}=a.

$$2016-\frac{8065+\sqrt{16143}i}{8}+\sqrt{\frac{8065+\sqrt{16143}i}{8}-2017}=\frac{8065+\sqrt{16143}i}{8}$$

Simplify. The value a=\frac{8065+\sqrt{16143}i}{8} satisfies the equation.

$$\frac{8065}{8}+\frac{1}{8}i\times 16143^{\frac{1}{2}}=\frac{8065}{8}+\frac{1}{8}i\times 16143^{\frac{1}{2}}$$

Equation \sqrt{a-2017}=2a-2016 has a unique solution.

$$a=\frac{8065+\sqrt{16143}i}{8}$$