This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — Here's the solution to question 2: i) Given the quadratic equation x^2 + (2k+2)x + 2k+5 = 0. Let the roots be and . From the equation, we have: Sum of roots: + = -(2k+2) Product of roots: = 2k+5 a) Find the values of the constant k for which = . If the roots are equal, the discriminant () must be zero. The discriminant is = b^2 - 4ac. Here, a=1, b=(2k+2), c=(2k+5). = (2k+2)^2 - 4(1)(2k+5) = 0 4k^2 + 8k + 4 - 8k - 20 = 0 4k^2 - 16 = 0 4k^2 = 16 k^2 = 4 k = ± sqrt(4) k = ± 2 The values of k are 2 and -2. k=2, -2 b) For k > 0, find the quadratic equation with integral coefficients, whose roots are 2 + and 2 + . Since k > 0, we use k=2. Substitute k=2 into the sum and product of roots for the original equation: + = -(2(2)+2) = -(4+2) = -6 = 2(2)+5 = 4+5 = 9 Let the new roots be r_1 = 2 + and r_2 = 2 + . Step 1: Calculate the sum of the new roots (r_1 + r_2). r_1 + r_2 = (2 + ) + (2 + ) r_1 + r_2 = 2 + 2 + 2 r_1 + r_2 = 2( + ) + 2 Substitute the values of + and : r_1 + r_2 = 2(-6) + 2(9) r_1 + r_2 = -12 + 18 r_1 + r_2 = 6 Step 2: Calculate the product of the new roots (r_1 r_2). r_1 r_2 = (2 + )(2 + ) Factor out from the first term and from the second term: r_1 r_2 = (2+) · (2+) r_1 r_2 = (2+)(2+) r_1 r_2 = (4 + 2 + 2 + ) r_1 r_2 = (4 + 2( + ) + ) Substitute the values of + and : r_1 r_2 = 9 (4 + 2(-6) + 9) r_1 r_2 = 9 (4 - 12 + 9) r_1 r_2 = 9 (1) r_1 r_2 = 9 Step 3: Form the quadratic equation. A quadratic equation with roots r_1 and r_2 is given by x^2 - (r_1+r_2)x + r_1r_2 = 0. x^2 - (6)x + 9 = 0 x^2 - 6x + 9 = 0 x^2 - 6x + 9 = 0 ii) Find the range of values of x for which (2-x)/(x+1) 0. Step 1: Find the critical points. The numerator is zero when 2-x=0 x=2. The denominator is zero when x+1=0 x=-1. These points divide the number line into three intervals: (-, -1), (-1, 2), and (2, ). Step 2: Test a value in each interval. • For x < -1 (e.g., x=-2): (2-(-2))/(-2+1) = (4)/(-1) = -4. This is < 0. • For -1 < x < 2 (e.g., x=0): (2-0)/(0+1) = (2)/(1) = 2. This is 0. • For x > 2 (e.g., x=3): (2-3)/(3+1) = (-1)/(4). This is < 0. Step 3: Determine the solution based on the inequality. We need (2-x)/(x+1) 0. This is satisfied when -1 < x < 2. Since the inequality includes "equal to" (), the value x=2 (where the numerator is zero) is included in the solution. The value x=-1 (where the denominator is zero) makes the expression undefined, so it must be excluded. Therefore, the range of values for x is -1 < x 2. -1 < x 2 What's next?