Solve (19,2)^2+b^2=24^2

$$( 19,2 ) ^ { 2 } + b ^ { 2 } = 24 ^ { 2 }$$

Solve for b

$$b=14,4$$

$$b=-14,4$$

Steps Using Difference of Squares

Calculate 19,2 to the power of 2 and get 368,64.

$$368,64+b^{2}=24^{2}$$

Calculate 24 to the power of 2 and get 576.

$$368,64+b^{2}=576$$

Subtract 576 from both sides.

$$368,64+b^{2}-576=0$$

Subtract 576 from 368,64 to get -207,36.

$$-207,36+b^{2}=0$$

Consider -207,36+b^{2}. Rewrite -207,36+b^{2} as b^{2}-\left(\frac{72}{5}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(b-\frac{72}{5}\right)\left(b+\frac{72}{5}\right)=0$$

To find equation solutions, solve b-\frac{72}{5}=0 and b+\frac{72}{5}=0.

$$b=\frac{72}{5} b=-\frac{72}{5}$$

Steps by Finding Square Root

Calculate 19,2 to the power of 2 and get 368,64.

$$368,64+b^{2}=24^{2}$$

Calculate 24 to the power of 2 and get 576.

$$368,64+b^{2}=576$$

Subtract 368,64 from both sides.

$$b^{2}=576-368,64$$

Subtract 368,64 from 576 to get 207,36.

$$b^{2}=207,36$$

Take the square root of both sides of the equation.

$$b=\frac{72}{5} b=-\frac{72}{5}$$

Steps Using the Quadratic Formula

Calculate 19,2 to the power of 2 and get 368,64.

$$368,64+b^{2}=24^{2}$$

Calculate 24 to the power of 2 and get 576.

$$368,64+b^{2}=576$$

Subtract 576 from both sides.

$$368,64+b^{2}-576=0$$

Subtract 576 from 368,64 to get -207,36.

$$-207,36+b^{2}=0$$

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.

$$b^{2}-207,36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -207,36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{0±\sqrt{0^{2}-4\left(-207,36\right)}}{2}$$

Square 0.

$$b=\frac{0±\sqrt{-4\left(-207,36\right)}}{2}$$

Multiply -4 times -207,36.

$$b=\frac{0±\sqrt{829,44}}{2}$$

Take the square root of 829,44.

$$b=\frac{0±\frac{144}{5}}{2}$$

Now solve the equation b=\frac{0±\frac{144}{5}}{2} when ± is plus.

$$b=\frac{72}{5}$$

Now solve the equation b=\frac{0±\frac{144}{5}}{2} when ± is minus.

$$b=-\frac{72}{5}$$

The equation is now solved.

$$b=\frac{72}{5} b=-\frac{72}{5}$$