Solve 12-a=frac{sqrt{a^2+10}}{2}

$$12 - a = \frac { \sqrt { a ^ { 2 } + 10 } } { 2 }$$

Solve for a

$$a=-\frac{\sqrt{606}}{3}+16\approx 7.794310917$$

Solution Steps

Multiply both sides of the equation by 2.

$$24-2a=\sqrt{a^{2}+10}$$

Subtract \sqrt{a^{2}+10} from both sides.

$$24-2a-\sqrt{a^{2}+10}=0$$

Subtract 24-2a from both sides of the equation.

$$-\sqrt{a^{2}+10}=-\left(24-2a\right)$$

Cancel out -1 on both sides.

$$\sqrt{a^{2}+10}=24-2a$$

Square both sides of the equation.

$$\left(\sqrt{a^{2}+10}\right)^{2}=\left(24-2a\right)^{2}$$

Calculate \sqrt{a^{2}+10} to the power of 2 and get a^{2}+10.

$$a^{2}+10=\left(24-2a\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(24-2a\right)^{2}.

$$a^{2}+10=576-96a+4a^{2}$$

Subtract 576 from both sides.

$$a^{2}+10-576=-96a+4a^{2}$$

Subtract 576 from 10 to get -566.

$$a^{2}-566=-96a+4a^{2}$$

Add 96a to both sides.

$$a^{2}-566+96a=4a^{2}$$

Subtract 4a^{2} from both sides.

$$a^{2}-566+96a-4a^{2}=0$$

Combine a^{2} and -4a^{2} to get -3a^{2}.

$$-3a^{2}-566+96a=0$$

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.

$$-3a^{2}+96a-566=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 96 for b, and -566 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-96±\sqrt{96^{2}-4\left(-3\right)\left(-566\right)}}{2\left(-3\right)}$$

Square 96.

$$a=\frac{-96±\sqrt{9216-4\left(-3\right)\left(-566\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$a=\frac{-96±\sqrt{9216+12\left(-566\right)}}{2\left(-3\right)}$$

Multiply 12 times -566.

$$a=\frac{-96±\sqrt{9216-6792}}{2\left(-3\right)}$$

Add 9216 to -6792.

$$a=\frac{-96±\sqrt{2424}}{2\left(-3\right)}$$

Take the square root of 2424.

$$a=\frac{-96±2\sqrt{606}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$a=\frac{-96±2\sqrt{606}}{-6}$$

Now solve the equation a=\frac{-96±2\sqrt{606}}{-6} when ± is plus. Add -96 to 2\sqrt{606}.

$$a=\frac{2\sqrt{606}-96}{-6}$$

Divide -96+2\sqrt{606} by -6.

$$a=-\frac{\sqrt{606}}{3}+16$$

Now solve the equation a=\frac{-96±2\sqrt{606}}{-6} when ± is minus. Subtract 2\sqrt{606} from -96.

$$a=\frac{-2\sqrt{606}-96}{-6}$$

Divide -96-2\sqrt{606} by -6.

$$a=\frac{\sqrt{606}}{3}+16$$

The equation is now solved.

$$a=-\frac{\sqrt{606}}{3}+16 a=\frac{\sqrt{606}}{3}+16$$

Substitute -\frac{\sqrt{606}}{3}+16 for a in the equation 12-a=\frac{\sqrt{a^{2}+10}}{2}.

$$12-\left(-\frac{\sqrt{606}}{3}+16\right)=\frac{\sqrt{\left(-\frac{\sqrt{606}}{3}+16\right)^{2}+10}}{2}$$

Simplify. The value a=-\frac{\sqrt{606}}{3}+16 satisfies the equation.

$$-4+\frac{1}{3}\times 606^{\frac{1}{2}}=\frac{1}{3}\times 606^{\frac{1}{2}}-4$$

Substitute \frac{\sqrt{606}}{3}+16 for a in the equation 12-a=\frac{\sqrt{a^{2}+10}}{2}.

$$12-\left(\frac{\sqrt{606}}{3}+16\right)=\frac{\sqrt{\left(\frac{\sqrt{606}}{3}+16\right)^{2}+10}}{2}$$

Simplify. The value a=\frac{\sqrt{606}}{3}+16 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-4-\frac{1}{3}\times 606^{\frac{1}{2}}=\frac{1}{3}\times 606^{\frac{1}{2}}+4$$

Equation \sqrt{a^{2}+10}=24-2a has a unique solution.

$$a=-\frac{\sqrt{606}}{3}+16$$