Solve (2-2cos36)x^2+(2+2cos36)*(x^2-1000/sin36)=2500

$$( 2 - 2 \cos 36 ) x ^ { 2 } + ( 2 + 2 \cos 36 ) \cdot ( x ^ { 2 } - \frac { 1000 } { \sin 36 } ) = 2500$$

Solve for x

$$x = \frac{75 \sqrt{13290470124050735325737974226149}}{5877852522924731} = 46\frac{3039511849886982}{5877852522924731} \approx 46.517112642$$

$$x = -\frac{75 \sqrt{13290470124050735325737974226149}}{5877852522924731} = -46\frac{3039511849886982}{5877852522924731} \approx -46.517112642$$

Steps by Finding Square Root

Evaluate trigonometric functions in the problem

$${(2 - 2 \cdot 0.8090169943749475)} x ^ {2} + {(2 + 2 \cdot 0.8090169943749475)} \cdot {(x ^ {2} - \frac{1000}{0.5877852522924731})} = 2500$$

Multiply 2 and 0.8090169943749475 to get 1.618033988749895.

$$\left(2-1.618033988749895\right)x^{2}+\left(2+2\times 0.8090169943749475\right)\left(x^{2}-\frac{1000}{0.5877852522924731}\right)=2500$$

Subtract 1.618033988749895 from 2 to get 0.381966011250105.

$$0.381966011250105x^{2}+\left(2+2\times 0.8090169943749475\right)\left(x^{2}-\frac{1000}{0.5877852522924731}\right)=2500$$

Multiply 2 and 0.8090169943749475 to get 1.618033988749895.

$$0.381966011250105x^{2}+\left(2+1.618033988749895\right)\left(x^{2}-\frac{1000}{0.5877852522924731}\right)=2500$$

Add 2 and 1.618033988749895 to get 3.618033988749895.

$$0.381966011250105x^{2}+3.618033988749895\left(x^{2}-\frac{1000}{0.5877852522924731}\right)=2500$$

Expand \frac{1000}{0.5877852522924731} by multiplying both numerator and the denominator by 10000000000000000.

$$0.381966011250105x^{2}+3.618033988749895\left(x^{2}-\frac{10000000000000000000}{5877852522924731}\right)=2500$$

Use the distributive property to multiply 3.618033988749895 by x^{2}-\frac{10000000000000000000}{5877852522924731}.

$$0.381966011250105x^{2}+3.618033988749895x^{2}-\frac{36180339887498950000}{5877852522924731}=2500$$

Combine 0.381966011250105x^{2} and 3.618033988749895x^{2} to get 4x^{2}.

$$4x^{2}-\frac{36180339887498950000}{5877852522924731}=2500$$

Add \frac{36180339887498950000}{5877852522924731} to both sides.

$$4x^{2}=2500+\frac{36180339887498950000}{5877852522924731}$$

Add 2500 and \frac{36180339887498950000}{5877852522924731} to get \frac{50874971194810777500}{5877852522924731}.

$$4x^{2}=\frac{50874971194810777500}{5877852522924731}$$

Divide both sides by 4.

$$x^{2}=\frac{\frac{50874971194810777500}{5877852522924731}}{4}$$

Express \frac{\frac{50874971194810777500}{5877852522924731}}{4} as a single fraction.

$$x^{2}=\frac{50874971194810777500}{5877852522924731\times 4}$$

Multiply 5877852522924731 and 4 to get 23511410091698924.

$$x^{2}=\frac{50874971194810777500}{23511410091698924}$$

Reduce the fraction \frac{50874971194810777500}{23511410091698924} to lowest terms by extracting and canceling out 4.

$$x^{2}=\frac{12718742798702694375}{5877852522924731}$$

Take the square root of both sides of the equation.

$$x=\frac{75\sqrt{13290470124050735325737974226149}}{5877852522924731} x=-\frac{75\sqrt{13290470124050735325737974226149}}{5877852522924731}$$

Steps Using the Quadratic Formula

Evaluate trigonometric functions in the problem

$${(2 - 2 \cdot 0.8090169943749475)} x ^ {2} + {(2 + 2 \cdot 0.8090169943749475)} \cdot {(x ^ {2} - \frac{1000}{0.5877852522924731})} = 2500$$

Multiply 2 and 0.8090169943749475 to get 1.618033988749895.

$$\left(2-1.618033988749895\right)x^{2}+\left(2+2\times 0.8090169943749475\right)\left(x^{2}-\frac{1000}{0.5877852522924731}\right)=2500$$

Subtract 1.618033988749895 from 2 to get 0.381966011250105.

$$0.381966011250105x^{2}+\left(2+2\times 0.8090169943749475\right)\left(x^{2}-\frac{1000}{0.5877852522924731}\right)=2500$$

Multiply 2 and 0.8090169943749475 to get 1.618033988749895.

$$0.381966011250105x^{2}+\left(2+1.618033988749895\right)\left(x^{2}-\frac{1000}{0.5877852522924731}\right)=2500$$

Add 2 and 1.618033988749895 to get 3.618033988749895.

$$0.381966011250105x^{2}+3.618033988749895\left(x^{2}-\frac{1000}{0.5877852522924731}\right)=2500$$

Expand \frac{1000}{0.5877852522924731} by multiplying both numerator and the denominator by 10000000000000000.

$$0.381966011250105x^{2}+3.618033988749895\left(x^{2}-\frac{10000000000000000000}{5877852522924731}\right)=2500$$

Use the distributive property to multiply 3.618033988749895 by x^{2}-\frac{10000000000000000000}{5877852522924731}.

$$0.381966011250105x^{2}+3.618033988749895x^{2}-\frac{36180339887498950000}{5877852522924731}=2500$$

Combine 0.381966011250105x^{2} and 3.618033988749895x^{2} to get 4x^{2}.

$$4x^{2}-\frac{36180339887498950000}{5877852522924731}=2500$$

Subtract 2500 from both sides.

$$4x^{2}-\frac{36180339887498950000}{5877852522924731}-2500=0$$

Subtract 2500 from -\frac{36180339887498950000}{5877852522924731} to get -\frac{50874971194810777500}{5877852522924731}.

$$4x^{2}-\frac{50874971194810777500}{5877852522924731}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 0 for b, and -\frac{50874971194810777500}{5877852522924731} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 4\left(-\frac{50874971194810777500}{5877852522924731}\right)}}{2\times 4}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 4\left(-\frac{50874971194810777500}{5877852522924731}\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{0±\sqrt{-16\left(-\frac{50874971194810777500}{5877852522924731}\right)}}{2\times 4}$$

Multiply -16 times -\frac{50874971194810777500}{5877852522924731}.

$$x=\frac{0±\sqrt{\frac{813999539116972440000}{5877852522924731}}}{2\times 4}$$

Take the square root of \frac{813999539116972440000}{5877852522924731}.

$$x=\frac{0±\frac{600\sqrt{13290470124050735325737974226149}}{5877852522924731}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{0±\frac{600\sqrt{13290470124050735325737974226149}}{5877852522924731}}{8}$$

Now solve the equation x=\frac{0±\frac{600\sqrt{13290470124050735325737974226149}}{5877852522924731}}{8} when ± is plus.

$$x=\frac{75\sqrt{13290470124050735325737974226149}}{5877852522924731}$$

Now solve the equation x=\frac{0±\frac{600\sqrt{13290470124050735325737974226149}}{5877852522924731}}{8} when ± is minus.

$$x=-\frac{75\sqrt{13290470124050735325737974226149}}{5877852522924731}$$

The equation is now solved.

$$x=\frac{75\sqrt{13290470124050735325737974226149}}{5877852522924731} x=-\frac{75\sqrt{13290470124050735325737974226149}}{5877852522924731}$$