Solve 113.6^2=F^2+2F^2-2(F)(2F)cos44

$$113.6 ^ { 2 } = F ^ { 2 } + 2 F ^ { 2 } - 2 ( F ) ( 2 F ) \cos 44$$

Solve for F

$$F = \frac{1420000000 \sqrt{19162624788343}}{19162624788343} = 324\frac{7378794640461}{19162624788343} \approx 324.385061792$$

$$F = -\frac{1420000000 \sqrt{19162624788343}}{19162624788343} = -324\frac{7378794640461}{19162624788343} \approx -324.385061792$$

Steps by Finding Square Root

Evaluate trigonometric functions in the problem

$$113.6 ^ {2} = F ^ {2} + 2 F ^ {2} - 2 {(F)} {(2 F)} 0.7193398003386512$$

Multiply F and F to get F^{2}.

$$113.6^{2}=F^{2}+2F^{2}-2F^{2}\times 2\times 0.7193398003386512$$

Calculate 113.6 to the power of 2 and get 12904.96.

$$12904.96=F^{2}+2F^{2}-2F^{2}\times 2\times 0.7193398003386512$$

Combine F^{2} and 2F^{2} to get 3F^{2}.

$$12904.96=3F^{2}-2F^{2}\times 2\times 0.7193398003386512$$

Multiply 2 and 2 to get 4.

$$12904.96=3F^{2}-4F^{2}\times 0.7193398003386512$$

Multiply 4 and 0.7193398003386512 to get 2.8773592013546048.

$$12904.96=3F^{2}-2.8773592013546048F^{2}$$

Combine 3F^{2} and -2.8773592013546048F^{2} to get 0.1226407986453952F^{2}.

$$12904.96=0.1226407986453952F^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$0.1226407986453952F^{2}=12904.96$$

Divide both sides by 0.1226407986453952.

$$F^{2}=\frac{12904.96}{0.1226407986453952}$$

Expand \frac{12904.96}{0.1226407986453952} by multiplying both numerator and the denominator by 10000000000000000.

$$F^{2}=\frac{129049600000000000000}{1226407986453952}$$

Reduce the fraction \frac{129049600000000000000}{1226407986453952} to lowest terms by extracting and canceling out 64.

$$F^{2}=\frac{2016400000000000000}{19162624788343}$$

Take the square root of both sides of the equation.

$$F=\frac{1420000000\sqrt{19162624788343}}{19162624788343} F=-\frac{1420000000\sqrt{19162624788343}}{19162624788343}$$

Steps Using the Quadratic Formula

Evaluate trigonometric functions in the problem

$$113.6 ^ {2} = F ^ {2} + 2 F ^ {2} - 2 {(F)} {(2 F)} 0.7193398003386512$$

Multiply F and F to get F^{2}.

$$113.6^{2}=F^{2}+2F^{2}-2F^{2}\times 2\times 0.7193398003386512$$

Calculate 113.6 to the power of 2 and get 12904.96.

$$12904.96=F^{2}+2F^{2}-2F^{2}\times 2\times 0.7193398003386512$$

Combine F^{2} and 2F^{2} to get 3F^{2}.

$$12904.96=3F^{2}-2F^{2}\times 2\times 0.7193398003386512$$

Multiply 2 and 2 to get 4.

$$12904.96=3F^{2}-4F^{2}\times 0.7193398003386512$$

Multiply 4 and 0.7193398003386512 to get 2.8773592013546048.

$$12904.96=3F^{2}-2.8773592013546048F^{2}$$

Combine 3F^{2} and -2.8773592013546048F^{2} to get 0.1226407986453952F^{2}.

$$12904.96=0.1226407986453952F^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$0.1226407986453952F^{2}=12904.96$$

Subtract 12904.96 from both sides.

$$0.1226407986453952F^{2}-12904.96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.1226407986453952 for a, 0 for b, and -12904.96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$F=\frac{0±\sqrt{0^{2}-4\times 0.1226407986453952\left(-12904.96\right)}}{2\times 0.1226407986453952}$$

Square 0.

$$F=\frac{0±\sqrt{-4\times 0.1226407986453952\left(-12904.96\right)}}{2\times 0.1226407986453952}$$

Multiply -4 times 0.1226407986453952.

$$F=\frac{0±\sqrt{-0.4905631945815808\left(-12904.96\right)}}{2\times 0.1226407986453952}$$

Multiply -0.4905631945815808 times -12904.96 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$F=\frac{0±\sqrt{6330.698403547516960768}}{2\times 0.1226407986453952}$$

Take the square root of 6330.698403547516960768.

$$F=\frac{0±\frac{71\sqrt{19162624788343}}{3906250}}{2\times 0.1226407986453952}$$

Multiply 2 times 0.1226407986453952.

$$F=\frac{0±\frac{71\sqrt{19162624788343}}{3906250}}{0.2452815972907904}$$

Now solve the equation F=\frac{0±\frac{71\sqrt{19162624788343}}{3906250}}{0.2452815972907904} when ± is plus.

$$F=\frac{1420000000\sqrt{19162624788343}}{19162624788343}$$

Now solve the equation F=\frac{0±\frac{71\sqrt{19162624788343}}{3906250}}{0.2452815972907904} when ± is minus.

$$F=-\frac{1420000000\sqrt{19162624788343}}{19162624788343}$$

The equation is now solved.

$$F=\frac{1420000000\sqrt{19162624788343}}{19162624788343} F=-\frac{1420000000\sqrt{19162624788343}}{19162624788343}$$