Solve left(78-100right)^2+y^2=100^2

$${ \left(78-100 \right) }^{ 2 } + { y }^{ 2 } = { 100 }^{ 2 }$$

Solve for y

$$y=2\sqrt{2379}\approx 97.549987186$$

$$y=-2\sqrt{2379}\approx -97.549987186$$

Steps by Finding Square Root

Subtract 100 from 78 to get -22.

$$\left(-22\right)^{2}+y^{2}=100^{2}$$

Calculate -22 to the power of 2 and get 484.

$$484+y^{2}=100^{2}$$

Calculate 100 to the power of 2 and get 10000.

$$484+y^{2}=10000$$

Subtract 484 from both sides.

$$y^{2}=10000-484$$

Subtract 484 from 10000 to get 9516.

$$y^{2}=9516$$

Take the square root of both sides of the equation.

$$y=2\sqrt{2379} y=-2\sqrt{2379}$$

Steps Using the Quadratic Formula

Subtract 100 from 78 to get -22.

$$\left(-22\right)^{2}+y^{2}=100^{2}$$

Calculate -22 to the power of 2 and get 484.

$$484+y^{2}=100^{2}$$

Calculate 100 to the power of 2 and get 10000.

$$484+y^{2}=10000$$

Subtract 10000 from both sides.

$$484+y^{2}-10000=0$$

Subtract 10000 from 484 to get -9516.

$$-9516+y^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$y^{2}-9516=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -9516 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\left(-9516\right)}}{2}$$

Square 0.

$$y=\frac{0±\sqrt{-4\left(-9516\right)}}{2}$$

Multiply -4 times -9516.

$$y=\frac{0±\sqrt{38064}}{2}$$

Take the square root of 38064.

$$y=\frac{0±4\sqrt{2379}}{2}$$

Now solve the equation y=\frac{0±4\sqrt{2379}}{2} when ± is plus.

$$y=2\sqrt{2379}$$

Now solve the equation y=\frac{0±4\sqrt{2379}}{2} when ± is minus.

$$y=-2\sqrt{2379}$$

The equation is now solved.

$$y=2\sqrt{2379} y=-2\sqrt{2379}$$