Solve sqrt{2(8+a)}=a

$$\sqrt { 2 ( 8 + a ) } = a$$

Solve for a

$$a=\sqrt{17}+1\approx 5.123105626$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{2\left(8+a\right)}\right)^{2}=a^{2}$$

Use the distributive property to multiply 2 by 8+a.

$$\left(\sqrt{16+2a}\right)^{2}=a^{2}$$

Calculate \sqrt{16+2a} to the power of 2 and get 16+2a.

$$16+2a=a^{2}$$

Subtract a^{2} from both sides.

$$16+2a-a^{2}=0$$

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.

$$-a^{2}+2a+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Square 2.

$$a=\frac{-2±\sqrt{4-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-2±\sqrt{4+4\times 16}}{2\left(-1\right)}$$

Multiply 4 times 16.

$$a=\frac{-2±\sqrt{4+64}}{2\left(-1\right)}$$

Add 4 to 64.

$$a=\frac{-2±\sqrt{68}}{2\left(-1\right)}$$

Take the square root of 68.

$$a=\frac{-2±2\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-2±2\sqrt{17}}{-2}$$

Now solve the equation a=\frac{-2±2\sqrt{17}}{-2} when ± is plus. Add -2 to 2\sqrt{17}.

$$a=\frac{2\sqrt{17}-2}{-2}$$

Divide -2+2\sqrt{17} by -2.

$$a=1-\sqrt{17}$$

Now solve the equation a=\frac{-2±2\sqrt{17}}{-2} when ± is minus. Subtract 2\sqrt{17} from -2.

$$a=\frac{-2\sqrt{17}-2}{-2}$$

Divide -2-2\sqrt{17} by -2.

$$a=\sqrt{17}+1$$

The equation is now solved.

$$a=1-\sqrt{17} a=\sqrt{17}+1$$

Substitute 1-\sqrt{17} for a in the equation \sqrt{2\left(8+a\right)}=a.

$$\sqrt{2\left(8+1-\sqrt{17}\right)}=1-\sqrt{17}$$

Simplify. The value a=1-\sqrt{17} does not satisfy the equation because the left and the right hand side have opposite signs.

$$17^{\frac{1}{2}}-1=1-17^{\frac{1}{2}}$$

Substitute \sqrt{17}+1 for a in the equation \sqrt{2\left(8+a\right)}=a.

$$\sqrt{2\left(8+\sqrt{17}+1\right)}=\sqrt{17}+1$$

Simplify. The value a=\sqrt{17}+1 satisfies the equation.

$$1+17^{\frac{1}{2}}=1+17^{\frac{1}{2}}$$

Equation \sqrt{2\left(a+8\right)}=a has a unique solution.

$$a=\sqrt{17}+1$$