The roots of equation $4{x^2} - 2\sqrt 5 \;x +1 = 0\;{\text{are}}$
- $\sin {36^ \circ };\sin {18^ \circ }$
- $\sin {18^ \circ };\cos {36^ \circ }$
- $\sin {36^ \circ };\cos {18^ \circ }$
- $\cos {18^ \circ };\cos {36^ \circ }$
Reveal answer
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B
Correct answer
Explanation
The equation is 4x^2 - 2*sqrt(5)*x + 1 = 0. Roots are (2*sqrt(5) +/- sqrt(20 - 16)) / 8 = (2*sqrt(5) +/- 2) / 8 = (sqrt(5) +/- 1) / 4. These values correspond to sin(72) and sin(18), or sin(18) and cos(36).
AI explanation
Using the quadratic formula, the roots of the equation 4x^2 - 2 * sqrt(5) * x + 1 = 0 are found by calculating (2 * sqrt(5) +/- sqrt(20 - 16)) / 8. This simplifies to (2 * sqrt(5) +/- 2) / 8, or (sqrt(5) +/- 1) / 4. Since the exact values of sin(18 degrees) and cos(36 degrees) are both (sqrt(5) - 1)/4, the roots are sin(18 degrees) and cos(36 degrees).