Solve left(34-100right)^2+y^2=100^2

$${ \left(34-100 \right) }^{ 2 } + { y }^{ 2 } = { 100 }^{ 2 }$$

Solve for y

$$y=2\sqrt{1411}\approx 75.126559884$$

$$y=-2\sqrt{1411}\approx -75.126559884$$

Steps by Finding Square Root

Subtract 100 from 34 to get -66.

$$\left(-66\right)^{2}+y^{2}=100^{2}$$

Calculate -66 to the power of 2 and get 4356.

$$4356+y^{2}=100^{2}$$

Calculate 100 to the power of 2 and get 10000.

$$4356+y^{2}=10000$$

Subtract 4356 from both sides.

$$y^{2}=10000-4356$$

Subtract 4356 from 10000 to get 5644.

$$y^{2}=5644$$

Take the square root of both sides of the equation.

$$y=2\sqrt{1411} y=-2\sqrt{1411}$$

Steps Using the Quadratic Formula

Subtract 100 from 34 to get -66.

$$\left(-66\right)^{2}+y^{2}=100^{2}$$

Calculate -66 to the power of 2 and get 4356.

$$4356+y^{2}=100^{2}$$

Calculate 100 to the power of 2 and get 10000.

$$4356+y^{2}=10000$$

Subtract 10000 from both sides.

$$4356+y^{2}-10000=0$$

Subtract 10000 from 4356 to get -5644.

$$-5644+y^{2}=0$$

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.

$$y^{2}-5644=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -5644 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\left(-5644\right)}}{2}$$

Square 0.

$$y=\frac{0±\sqrt{-4\left(-5644\right)}}{2}$$

Multiply -4 times -5644.

$$y=\frac{0±\sqrt{22576}}{2}$$

Take the square root of 22576.

$$y=\frac{0±4\sqrt{1411}}{2}$$

Now solve the equation y=\frac{0±4\sqrt{1411}}{2} when ± is plus.

$$y=2\sqrt{1411}$$

Now solve the equation y=\frac{0±4\sqrt{1411}}{2} when ± is minus.

$$y=-2\sqrt{1411}$$

The equation is now solved.

$$y=2\sqrt{1411} y=-2\sqrt{1411}$$