Given P = 4 + sqrt(2) and Q = 2 + sqrt(2).

Here are the solutions to the problems: Problem 1: Given P = 4 + sqrt(2) and Q = 2 + sqrt(2). We need to find a, b, c such that (P)/(Q) = a + bsqrt(c). Step 1: Substitute the values of P and Q into the expression (P)/(Q). (P)/(Q) = 4 + sqrt(2)2 + sqrt(2) Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2 - sqrt(2). (P)/(Q) = 4 + sqrt(2)2 + sqrt(2) × 2 - sqrt(2)2 - sqrt(2) Step 3: Expand the numerator and the denominator. Numerator: (4 + sqrt(2))(2 - sqrt(2)) = 4(2) - 4sqrt(2) + 2sqrt(2) - (sqrt(2))^2 = 8 - 4sqrt(2) + 2sqrt(2) - 2 = 6 - 2sqrt(2) Denominator: (2 + sqrt(2))(2 - sqrt(2)) = 2^2 - (sqrt(2))^2 = 4 - 2 = 2 Step 4: Combine the simplified numerator and denominator. (P)/(Q) = 6 - 2sqrt(2)2 Step 5: Simplify the fraction. (P)/(Q) = (6)/(2) - 2sqrt(2)2 (P)/(Q) = 3 - sqrt(2) Step 6: Compare this result with a + bsqrt(c). 3 - sqrt(2) = a + bsqrt(c) By comparison, we have: a = 3 b = -1 c = 2 The values are a = 3, b = -1, c = 2. --- Problem 2: Solve the equation 4^2 x + 4 x = 5 for 0^ x 360^. Step 1: Use the trigonometric identity ^2 x + ^2 x = 1, which implies ^2 x = 1 - ^2 x. Substitute this into the equation. 4(1 - ^2 x) + 4 x = 5 Step 2: Expand and rearrange the equation into a standard quadratic form A^2 x + B x + C = 0. 4 - 4^2 x + 4 x = 5 -4^2 x + 4 x + 4 - 5 = 0 -4^2 x + 4 x - 1 = 0 Multiply by -1 to make the leading coefficient positive: 4^2 x - 4 x + 1 = 0 Step 3: Let y = x. The equation becomes a quadratic equation in y: 4y^2 - 4y + 1 = 0 This is a perfect square trinomial: (2y - 1)^2 = 0. Alternatively, use the quadratic formula: y = -(-4) ± sqrt((-4)^2 - 4(4)(1))2(4) y = 4 ± sqrt(16 - 16)8 y = (4 ± 0)/(8) y = (4)/(8) y = (1)/(2) Step 4: Substitute back y = x. x = (1)/(2) Step 5: Find the values of x in the range 0^ x 360^ for which x = (1)/(2). The cosine function is positive in the first and fourth quadrants. The reference angle is ^-1((1)/(2)) = 60^. In the first quadrant: x = 60^ In the fourth quadrant: x = 360^ - 60^ = 300^ The solutions are x = 60^, 300^. --- Problem 3: Make a the subject of the formula: R = hsqrt(y^2 - a^2). Step 1: Divide both sides by h to isolate the square root term. (R)/(h) = sqrt(y^2 - a^2) Step 2: Square both sides of the equation to eliminate the square root. ((R)/(h))^2 = (sqrt(y^2 - a^2))^2 (R^2)/(h^2) = y^2 - a^2 Step 3: Rearrange the equation to isolate a^2. Add a^2 to both sides and subtract (R^2)/(h^2) from both sides. a^2 = y^2 - (R^2)/(h^2) Step 4: Take the square root of both sides to solve for a. Remember to include both positive and negative roots. a = ±sqrt(y^2 - (R^2)/(h^2)) This can also be written with a common denominator inside the square root: a = ±sqrt((y^2h^2 - R^2)/(h^2)) a = ±sqrt(y^2h^2 - R^2)h The subject of the formula is a = ±sqrt(y^2 - (R^2)/(h^2)) or a = ±sqrt(y^2h^2 - R^2)h.