Solve x^2=80^2+78^2-2*80*78cos(40)

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

Solve for x

$$x = \frac{\sqrt{4568383359179929}}{1250000} \approx 54.071853583$$

$$x = -\frac{\sqrt{4568383359179929}}{1250000} \approx -54.071853583$$

Steps by Finding Square Root

Evaluate trigonometric functions in the problem

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

Calculate 80 to the power of 2 and get 6400.

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

Calculate 78 to the power of 2 and get 6084.

$$x^{2}=6400+6084-2\times 80\times 78\times 0.766044443118978$$

Add 6400 and 6084 to get 12484.

$$x^{2}=12484-2\times 80\times 78\times 0.766044443118978$$

Multiply 2 and 80 to get 160.

$$x^{2}=12484-160\times 78\times 0.766044443118978$$

Multiply 160 and 78 to get 12480.

$$x^{2}=12484-12480\times 0.766044443118978$$

Multiply 12480 and 0.766044443118978 to get 9560.23465012484544.

$$x^{2}=12484-9560.23465012484544$$

Subtract 9560.23465012484544 from 12484 to get 2923.76534987515456.

$$x^{2}=2923.76534987515456$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{4568383359179929}}{1250000} x=-\frac{\sqrt{4568383359179929}}{1250000}$$

Steps Using the Quadratic Formula

Evaluate trigonometric functions in the problem

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

Calculate 80 to the power of 2 and get 6400.

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

Calculate 78 to the power of 2 and get 6084.

$$x^{2}=6400+6084-2\times 80\times 78\times 0.766044443118978$$

Add 6400 and 6084 to get 12484.

$$x^{2}=12484-2\times 80\times 78\times 0.766044443118978$$

Multiply 2 and 80 to get 160.

$$x^{2}=12484-160\times 78\times 0.766044443118978$$

Multiply 160 and 78 to get 12480.

$$x^{2}=12484-12480\times 0.766044443118978$$

Multiply 12480 and 0.766044443118978 to get 9560.23465012484544.

$$x^{2}=12484-9560.23465012484544$$

Subtract 9560.23465012484544 from 12484 to get 2923.76534987515456.

$$x^{2}=2923.76534987515456$$

Subtract 2923.76534987515456 from both sides.

$$x^{2}-2923.76534987515456=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -2923.76534987515456 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-2923.76534987515456\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-2923.76534987515456\right)}}{2}$$

Multiply -4 times -2923.76534987515456.

$$x=\frac{0±\sqrt{11695.06139950061824}}{2}$$

Take the square root of 11695.06139950061824.

$$x=\frac{0±\frac{\sqrt{4568383359179929}}{625000}}{2}$$

Now solve the equation x=\frac{0±\frac{\sqrt{4568383359179929}}{625000}}{2} when ± is plus.

$$x=\frac{\sqrt{4568383359179929}}{1250000}$$

Now solve the equation x=\frac{0±\frac{\sqrt{4568383359179929}}{625000}}{2} when ± is minus.

$$x=-\frac{\sqrt{4568383359179929}}{1250000}$$

The equation is now solved.

$$x=\frac{\sqrt{4568383359179929}}{1250000} x=-\frac{\sqrt{4568383359179929}}{1250000}$$