Solve sqrt{z}-7=sqrt{z-105}

$$\sqrt { z } - 7 = \sqrt { z - 105 }$$

Solve for z

$$z=121$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{z}-7\right)^{2}=\left(\sqrt{z-105}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{z}-7\right)^{2}.

$$\left(\sqrt{z}\right)^{2}-14\sqrt{z}+49=\left(\sqrt{z-105}\right)^{2}$$

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

$$z-14\sqrt{z}+49=\left(\sqrt{z-105}\right)^{2}$$

Calculate \sqrt{z-105} to the power of 2 and get z-105.

$$z-14\sqrt{z}+49=z-105$$

Subtract z from both sides.

$$z-14\sqrt{z}+49-z=-105$$

Combine z and -z to get 0.

$$-14\sqrt{z}+49=-105$$

Subtract 49 from both sides.

$$-14\sqrt{z}=-105-49$$

Subtract 49 from -105 to get -154.

$$-14\sqrt{z}=-154$$

Divide both sides by -14.

$$\sqrt{z}=\frac{-154}{-14}$$

Divide -154 by -14 to get 11.

$$\sqrt{z}=11$$

Square both sides of the equation.

$$z=121$$

Substitute 121 for z in the equation \sqrt{z}-7=\sqrt{z-105}.

$$\sqrt{121}-7=\sqrt{121-105}$$

Simplify. The value z=121 satisfies the equation.

$$4=4$$

Equation \sqrt{z}-7=\sqrt{z-105} has a unique solution.

$$z=121$$