Solve left(77-100right)^2+y^2=100^2

$${ \left(77-100 \right) }^{ 2 } + { y }^{ 2 } = { 100 }^{ 2 }$$

Solve for y

$$y=\sqrt{9471}\approx 97.319062881$$

$$y=-\sqrt{9471}\approx -97.319062881$$

Steps by Finding Square Root

Subtract 100 from 77 to get -23.

$$\left(-23\right)^{2}+y^{2}=100^{2}$$

Calculate -23 to the power of 2 and get 529.

$$529+y^{2}=100^{2}$$

Calculate 100 to the power of 2 and get 10000.

$$529+y^{2}=10000$$

Subtract 529 from both sides.

$$y^{2}=10000-529$$

Subtract 529 from 10000 to get 9471.

$$y^{2}=9471$$

Take the square root of both sides of the equation.

$$y=\sqrt{9471} y=-\sqrt{9471}$$

Steps Using the Quadratic Formula

Subtract 100 from 77 to get -23.

$$\left(-23\right)^{2}+y^{2}=100^{2}$$

Calculate -23 to the power of 2 and get 529.

$$529+y^{2}=100^{2}$$

Calculate 100 to the power of 2 and get 10000.

$$529+y^{2}=10000$$

Subtract 10000 from both sides.

$$529+y^{2}-10000=0$$

Subtract 10000 from 529 to get -9471.

$$-9471+y^{2}=0$$

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.

$$y^{2}-9471=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -9471 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\left(-9471\right)}}{2}$$

Square 0.

$$y=\frac{0±\sqrt{-4\left(-9471\right)}}{2}$$

Multiply -4 times -9471.

$$y=\frac{0±\sqrt{37884}}{2}$$

Take the square root of 37884.

$$y=\frac{0±2\sqrt{9471}}{2}$$

Now solve the equation y=\frac{0±2\sqrt{9471}}{2} when ± is plus.

$$y=\sqrt{9471}$$

Now solve the equation y=\frac{0±2\sqrt{9471}}{2} when ± is minus.

$$y=-\sqrt{9471}$$

The equation is now solved.

$$y=\sqrt{9471} y=-\sqrt{9471}$$