Solve 75^2+x^2=85^2

$$75 ^ { 2 } + x ^ { 2 } = 85 ^ { 2 }$$

Solve for x

$$x=-40$$

$$x=40$$

Steps Using Difference of Squares

Calculate 75 to the power of 2 and get 5625.

$$5625+x^{2}=85^{2}$$

Calculate 85 to the power of 2 and get 7225.

$$5625+x^{2}=7225$$

Subtract 7225 from both sides.

$$5625+x^{2}-7225=0$$

Subtract 7225 from 5625 to get -1600.

$$-1600+x^{2}=0$$

Consider -1600+x^{2}. Rewrite -1600+x^{2} as x^{2}-40^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-40\right)\left(x+40\right)=0$$

To find equation solutions, solve x-40=0 and x+40=0.

$$x=40 x=-40$$

Steps by Finding Square Root

Calculate 75 to the power of 2 and get 5625.

$$5625+x^{2}=85^{2}$$

Calculate 85 to the power of 2 and get 7225.

$$5625+x^{2}=7225$$

Subtract 5625 from both sides.

$$x^{2}=7225-5625$$

Subtract 5625 from 7225 to get 1600.

$$x^{2}=1600$$

Take the square root of both sides of the equation.

$$x=40 x=-40$$

Steps Using the Quadratic Formula

Calculate 75 to the power of 2 and get 5625.

$$5625+x^{2}=85^{2}$$

Calculate 85 to the power of 2 and get 7225.

$$5625+x^{2}=7225$$

Subtract 7225 from both sides.

$$5625+x^{2}-7225=0$$

Subtract 7225 from 5625 to get -1600.

$$-1600+x^{2}=0$$

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.

$$x^{2}-1600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -1600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-1600\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1600\right)}}{2}$$

Multiply -4 times -1600.

$$x=\frac{0±\sqrt{6400}}{2}$$

Take the square root of 6400.

$$x=\frac{0±80}{2}$$

Now solve the equation x=\frac{0±80}{2} when ± is plus. Divide 80 by 2.

$$x=40$$

Now solve the equation x=\frac{0±80}{2} when ± is minus. Divide -80 by 2.

$$x=-40$$

The equation is now solved.

$$x=40 x=-40$$