Solve sqrt{a^2-4a+20}=sqrt{a}

$$\sqrt { a ^ { 2 } - 4 a + 20 } = \sqrt { a }$$

Solve for a

$$a=2\sqrt{5}e^{\arctan(\frac{\sqrt{55}}{5})i}\approx 2.5+3.708099244i$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{a^{2}-4a+20}\right)^{2}=\left(\sqrt{a}\right)^{2}$$

Calculate \sqrt{a^{2}-4a+20} to the power of 2 and get a^{2}-4a+20.

$$a^{2}-4a+20=\left(\sqrt{a}\right)^{2}$$

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

$$a^{2}-4a+20=a$$

Subtract a from both sides.

$$a^{2}-4a+20-a=0$$

Combine -4a and -a to get -5a.

$$a^{2}-5a+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 20}}{2}$$

Square -5.

$$a=\frac{-\left(-5\right)±\sqrt{25-4\times 20}}{2}$$

Multiply -4 times 20.

$$a=\frac{-\left(-5\right)±\sqrt{25-80}}{2}$$

Add 25 to -80.

$$a=\frac{-\left(-5\right)±\sqrt{-55}}{2}$$

Take the square root of -55.

$$a=\frac{-\left(-5\right)±\sqrt{55}i}{2}$$

The opposite of -5 is 5.

$$a=\frac{5±\sqrt{55}i}{2}$$

Now solve the equation a=\frac{5±\sqrt{55}i}{2} when ± is plus. Add 5 to i\sqrt{55}.

$$a=\frac{5+\sqrt{55}i}{2}$$

Now solve the equation a=\frac{5±\sqrt{55}i}{2} when ± is minus. Subtract i\sqrt{55} from 5.

$$a=\frac{-\sqrt{55}i+5}{2}$$

The equation is now solved.

$$a=\frac{5+\sqrt{55}i}{2} a=\frac{-\sqrt{55}i+5}{2}$$

Substitute \frac{5+\sqrt{55}i}{2} for a in the equation \sqrt{a^{2}-4a+20}=\sqrt{a}.

$$\sqrt{\left(\frac{5+\sqrt{55}i}{2}\right)^{2}-4\times \frac{5+\sqrt{55}i}{2}+20}=\sqrt{\frac{5+\sqrt{55}i}{2}}$$

Simplify. The value a=\frac{5+\sqrt{55}i}{2} satisfies the equation.

$$\frac{1}{2}\left(10+2i\times 55^{\frac{1}{2}}\right)^{\frac{1}{2}}=\left(\frac{5}{2}+\frac{1}{2}i\times 55^{\frac{1}{2}}\right)^{\frac{1}{2}}$$

Substitute \frac{-\sqrt{55}i+5}{2} for a in the equation \sqrt{a^{2}-4a+20}=\sqrt{a}.

$$\sqrt{\left(\frac{-\sqrt{55}i+5}{2}\right)^{2}-4\times \frac{-\sqrt{55}i+5}{2}+20}=\sqrt{\frac{-\sqrt{55}i+5}{2}}$$

Simplify. The value a=\frac{-\sqrt{55}i+5}{2} satisfies the equation.

$$\frac{1}{2}\left(10-2i\times 55^{\frac{1}{2}}\right)^{\frac{1}{2}}=\left(-\frac{1}{2}i\times 55^{\frac{1}{2}}+\frac{5}{2}\right)^{\frac{1}{2}}$$

List all solutions of \sqrt{a^{2}-4a+20}=\sqrt{a}.

$$a=\frac{5+\sqrt{55}i}{2} a=\frac{-\sqrt{55}i+5}{2}$$