Solve 2016-a+sqrt{a-207}=a

$$2016 - a + \sqrt { a - 207 } = a$$

Solve for a

$$a = \frac{\sqrt{12817} + 8065}{8} \approx 1022.27652377$$

Solution Steps

Subtract 2016-a from both sides of the equation.

$$\sqrt{a-207}=a-\left(2016-a\right)$$

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

$$\sqrt{a-207}=a-2016-\left(-a\right)$$

The opposite of -a is a.

$$\sqrt{a-207}=a-2016+a$$

Combine a and a to get 2a.

$$\sqrt{a-207}=2a-2016$$

Square both sides of the equation.

$$\left(\sqrt{a-207}\right)^{2}=\left(2a-2016\right)^{2}$$

Calculate \sqrt{a-207} to the power of 2 and get a-207.

$$a-207=\left(2a-2016\right)^{2}$$

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

$$a-207=4a^{2}-8064a+4064256$$

Subtract 4a^{2} from both sides.

$$a-207-4a^{2}=-8064a+4064256$$

Add 8064a to both sides.

$$a-207-4a^{2}+8064a=4064256$$

Combine a and 8064a to get 8065a.

$$8065a-207-4a^{2}=4064256$$

Subtract 4064256 from both sides.

$$8065a-207-4a^{2}-4064256=0$$

Subtract 4064256 from -207 to get -4064463.

$$8065a-4064463-4a^{2}=0$$

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.

$$-4a^{2}+8065a-4064463=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 8065 for b, and -4064463 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-8065±\sqrt{8065^{2}-4\left(-4\right)\left(-4064463\right)}}{2\left(-4\right)}$$

Square 8065.

$$a=\frac{-8065±\sqrt{65044225-4\left(-4\right)\left(-4064463\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$a=\frac{-8065±\sqrt{65044225+16\left(-4064463\right)}}{2\left(-4\right)}$$

Multiply 16 times -4064463.

$$a=\frac{-8065±\sqrt{65044225-65031408}}{2\left(-4\right)}$$

Add 65044225 to -65031408.

$$a=\frac{-8065±\sqrt{12817}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$a=\frac{-8065±\sqrt{12817}}{-8}$$

Now solve the equation a=\frac{-8065±\sqrt{12817}}{-8} when ± is plus. Add -8065 to \sqrt{12817}.

$$a=\frac{\sqrt{12817}-8065}{-8}$$

Divide -8065+\sqrt{12817} by -8.

$$a=\frac{8065-\sqrt{12817}}{8}$$

Now solve the equation a=\frac{-8065±\sqrt{12817}}{-8} when ± is minus. Subtract \sqrt{12817} from -8065.

$$a=\frac{-\sqrt{12817}-8065}{-8}$$

Divide -8065-\sqrt{12817} by -8.

$$a=\frac{\sqrt{12817}+8065}{8}$$

The equation is now solved.

$$a=\frac{8065-\sqrt{12817}}{8} a=\frac{\sqrt{12817}+8065}{8}$$

Substitute \frac{8065-\sqrt{12817}}{8} for a in the equation 2016-a+\sqrt{a-207}=a.

$$2016-\frac{8065-\sqrt{12817}}{8}+\sqrt{\frac{8065-\sqrt{12817}}{8}-207}=\frac{8065-\sqrt{12817}}{8}$$

Simplify. The value a=\frac{8065-\sqrt{12817}}{8} does not satisfy the equation.

$$\frac{8061}{8}+\frac{3}{8}\times 12817^{\frac{1}{2}}=\frac{8065}{8}-\frac{1}{8}\times 12817^{\frac{1}{2}}$$

Substitute \frac{\sqrt{12817}+8065}{8} for a in the equation 2016-a+\sqrt{a-207}=a.

$$2016-\frac{\sqrt{12817}+8065}{8}+\sqrt{\frac{\sqrt{12817}+8065}{8}-207}=\frac{\sqrt{12817}+8065}{8}$$

Simplify. The value a=\frac{\sqrt{12817}+8065}{8} satisfies the equation.

$$\frac{8065}{8}+\frac{1}{8}\times 12817^{\frac{1}{2}}=\frac{1}{8}\times 12817^{\frac{1}{2}}+\frac{8065}{8}$$

Equation \sqrt{a-207}=2a-2016 has a unique solution.

$$a=\frac{\sqrt{12817}+8065}{8}$$