Solve x^2=80^2+78^2-2(80)(18)cos40

$$x ^ { 2 } = 80 ^ { 2 } + 78 ^ { 2 } - 2 ( 80 ) ( 18 ) \cos 40$$

Solve for x

$$x = \frac{\sqrt{16059050005964599}}{1250000} \approx 101.379445667$$

$$x = -\frac{\sqrt{16059050005964599}}{1250000} \approx -101.379445667$$

Steps by Finding Square Root

Evaluate trigonometric functions in the problem

$$x ^ {2} = 80 ^ {2} + 78 ^ {2} - 2 {(80)} {(18)} \cdot 0.766044443118978$$

Calculate 80 to the power of 2 and get 6400.

$$x^{2}=6400+78^{2}-2\times 80\times 18\times 0.766044443118978$$

Calculate 78 to the power of 2 and get 6084.

$$x^{2}=6400+6084-2\times 80\times 18\times 0.766044443118978$$

Add 6400 and 6084 to get 12484.

$$x^{2}=12484-2\times 80\times 18\times 0.766044443118978$$

Multiply 2 and 80 to get 160.

$$x^{2}=12484-160\times 18\times 0.766044443118978$$

Multiply 160 and 18 to get 2880.

$$x^{2}=12484-2880\times 0.766044443118978$$

Multiply 2880 and 0.766044443118978 to get 2206.20799618265664.

$$x^{2}=12484-2206.20799618265664$$

Subtract 2206.20799618265664 from 12484 to get 10277.79200381734336.

$$x^{2}=10277.79200381734336$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{16059050005964599}}{1250000} x=-\frac{\sqrt{16059050005964599}}{1250000}$$

Steps Using the Quadratic Formula

Evaluate trigonometric functions in the problem

$$x ^ {2} = 80 ^ {2} + 78 ^ {2} - 2 {(80)} {(18)} \cdot 0.766044443118978$$

Calculate 80 to the power of 2 and get 6400.

$$x^{2}=6400+78^{2}-2\times 80\times 18\times 0.766044443118978$$

Calculate 78 to the power of 2 and get 6084.

$$x^{2}=6400+6084-2\times 80\times 18\times 0.766044443118978$$

Add 6400 and 6084 to get 12484.

$$x^{2}=12484-2\times 80\times 18\times 0.766044443118978$$

Multiply 2 and 80 to get 160.

$$x^{2}=12484-160\times 18\times 0.766044443118978$$

Multiply 160 and 18 to get 2880.

$$x^{2}=12484-2880\times 0.766044443118978$$

Multiply 2880 and 0.766044443118978 to get 2206.20799618265664.

$$x^{2}=12484-2206.20799618265664$$

Subtract 2206.20799618265664 from 12484 to get 10277.79200381734336.

$$x^{2}=10277.79200381734336$$

Subtract 10277.79200381734336 from both sides.

$$x^{2}-10277.79200381734336=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -10277.79200381734336 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-10277.79200381734336\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-10277.79200381734336\right)}}{2}$$

Multiply -4 times -10277.79200381734336.

$$x=\frac{0±\sqrt{41111.16801526937344}}{2}$$

Take the square root of 41111.16801526937344.

$$x=\frac{0±\frac{\sqrt{16059050005964599}}{625000}}{2}$$

Now solve the equation x=\frac{0±\frac{\sqrt{16059050005964599}}{625000}}{2} when ± is plus.

$$x=\frac{\sqrt{16059050005964599}}{1250000}$$

Now solve the equation x=\frac{0±\frac{\sqrt{16059050005964599}}{625000}}{2} when ± is minus.

$$x=-\frac{\sqrt{16059050005964599}}{1250000}$$

The equation is now solved.

$$x=\frac{\sqrt{16059050005964599}}{1250000} x=-\frac{\sqrt{16059050005964599}}{1250000}$$