If 1/(3-"√"5) - (2+"√"5)/(3+"√"5) = a + b"√"c, find the value of a, b and c.

Step 1: Rationalize the first term (1)/(3-sqrt(5)). To rationalize the denominator, multiply the numerator and denominator by the conjugate of 3-sqrt(5), which is 3+sqrt(5). (1)/(3-sqrt(5)) = (1)/(3-sqrt(5)) × 3+sqrt(5)3+sqrt(5) = 3+sqrt(5)(3)^2 - (sqrt(5))^2 = 3+sqrt(5)9 - 5 = 3+sqrt(5)4 Step 2: Rationalize the second term 2+2sqrt(5)3+sqrt(5). To rationalize the denominator, multiply the numerator and denominator by the conjugate of 3+sqrt(5), which is 3-sqrt(5). 2+2sqrt(5)3+sqrt(5) = 2+2sqrt(5)3+sqrt(5) × 3-sqrt(5)3-sqrt(5) = (2+2sqrt(5))(3-sqrt(5))(3)^2 - (sqrt(5))^2 Expand the numerator: (2+2sqrt(5))(3-sqrt(5)) = 2(3) - 2(sqrt(5)) + 2sqrt(5)(3) - 2sqrt(5)(sqrt(5)) = 6 - 2sqrt(5) + 6sqrt(5) - 2(5) = 6 + 4sqrt(5) - 10 = -4 + 4sqrt(5) The denominator is 9-5=4. So, the second term becomes: -4+4sqrt(5)4 = 4(-1+sqrt(5))4 = -1+sqrt(5) Step 3: Substitute the simplified terms back into the original equation. 3+sqrt(5)4 - (-1+sqrt(5)) = a+bsqrt(c) 3+sqrt(5)4 + 1 - sqrt(5) = a+bsqrt(c) Step 4: Combine the terms on the left-hand side. Find a common denominator, which is 4. 3+sqrt(5)4 + (4)/(4) - 4sqrt(5)4 = a+bsqrt(c) 3+sqrt(5) + 4 - 4sqrt(5)4 = a+bsqrt(c) Combine the rational parts and the irrational parts: (3+4) + (sqrt(5) - 4sqrt(5))4 = a+bsqrt(c) 7 - 3sqrt(5)4 = a+bsqrt(c) Step 5: Separate the rational and irrational parts to match the form a+bsqrt(c). (7)/(4) - 3sqrt(5)4 = a+bsqrt(c) This can be written as: (7)/(4) + (-(3)/(4))sqrt(5) = a+bsqrt(c) Step 6: Compare the terms to find the values of a, b, and c. By comparing the rational parts: a = (7)/(4) By comparing the coefficients of sqrt(5): b = -(3)/(4) By comparing the radicands: c = 5 The values are: a = (7)/(4) b = -(3)/(4) c = 5