Solve the quadratic equation 4 - 3x - 2x2 = 0 by completing the square.

Step 1: Rearrange the equation into the standard form ax^2 + bx + c = 0. 4 - 3x - 2x^2 = 0 -2x^2 - 3x + 4 = 0 Multiply by -1 to make the leading coefficient positive: 2x^2 + 3x - 4 = 0 Step 2: Divide the entire equation by the coefficient of x^2, which is 2: x^2 + (3)/(2)x - 2 = 0 Step 3: Move the constant term to the right side of the equation: x^2 + (3)/(2)x = 2 Step 4: Complete the square on the left side. Take half of the coefficient of x, square it, and add it to both sides. The coefficient of x is (3)/(2). Half of it is (1)/(2) × (3)/(2) = (3)/(4). Squaring this gives ((3)/(4))^2 = (9)/(16). Add (9)/(16) to both sides: x^2 + (3)/(2)x + (9)/(16) = 2 + (9)/(16) Step 5: Factor the left side as a perfect square and simplify the right side: (x + (3)/(4))^2 = (32)/(16) + (9)/(16) (x + (3)/(4))^2 = (41)/(16) Step 6: Take the square root of both sides: x + (3)/(4) = ±sqrt((41)/(16)) x + (3)/(4) = ±sqrt(41)4 Step 7: Solve for x: x = -(3)/(4) ± sqrt(41)4 x = -3 ± sqrt(41)4 The final answer is x = -3 ± sqrt(41)4. --- Step 1: Substitute the given 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 the 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))(sqrt(2)) = 8 - 4sqrt(2) + 2sqrt(2) - 2 = 6 - 2sqrt(2) Denominator (using the difference of squares formula (A+B)(A-B) = A^2 - B^2): (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 expression by dividing both terms in the numerator by 2: (P)/(Q) = (6)/(2) - 2sqrt(2)2 (P)/(Q) = 3 - sqrt(2) Step 6: Compare this result with the given form a + bsqrt(c): 3 - sqrt(2) = a + bsqrt(c) By comparing the terms, we find: a = 3 b = -1 c = 2 The final answer is a = 3, b = -1, c = 2. --- Step 1: Identify the given temperature readings and their precision. P = 22.5^ C Q = 19.4^ C