Solve 12-x+sqrt{12^2+x^2}=20

$$12-x+ \sqrt{ { 12 }^{ 2 } + { x }^{ 2 } } = 20$$

Solve for x

$$x=5$$

Solution Steps

Subtract 12-x from both sides of the equation.

$$\sqrt{12^{2}+x^{2}}=20-\left(12-x\right)$$

Calculate 12 to the power of 2 and get 144.

$$\sqrt{144+x^{2}}=20-\left(12-x\right)$$

To find the opposite of 12-x, find the opposite of each term.

$$\sqrt{144+x^{2}}=20-12+x$$

Subtract 12 from 20 to get 8.

$$\sqrt{144+x^{2}}=8+x$$

Square both sides of the equation.

$$\left(\sqrt{144+x^{2}}\right)^{2}=\left(8+x\right)^{2}$$

Calculate \sqrt{144+x^{2}} to the power of 2 and get 144+x^{2}.

$$144+x^{2}=\left(8+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8+x\right)^{2}.

$$144+x^{2}=64+16x+x^{2}$$

Subtract 16x from both sides.

$$144+x^{2}-16x=64+x^{2}$$

Subtract x^{2} from both sides.

$$144+x^{2}-16x-x^{2}=64$$

Combine x^{2} and -x^{2} to get 0.

$$144-16x=64$$

Subtract 144 from both sides.

$$-16x=64-144$$

Subtract 144 from 64 to get -80.

$$-16x=-80$$

Divide both sides by -16.

$$x=\frac{-80}{-16}$$

Divide -80 by -16 to get 5.

$$x=5$$

Substitute 5 for x in the equation 12-x+\sqrt{12^{2}+x^{2}}=20.

$$12-5+\sqrt{12^{2}+5^{2}}=20$$

Simplify. The value x=5 satisfies the equation.

$$20=20$$

Equation \sqrt{x^{2}+144}=x+8 has a unique solution.

$$x=5$$