Simplify (y2+y-6)/(y2-4) + (y+3)/(2y2+y-1)

Here are the solutions to Question 56. Question 56: Part (i): Simplify (y^2+y-6)/(y^2-4) + (y+3)/(2y^2+y-1) Step 1: Factorize the numerators and denominators of both fractions. For the first fraction: Numerator: y^2+y-6 = (y+3)(y-2) Denominator: y^2-4 = (y-2)(y+2) (difference of squares) For the second fraction: Numerator: y+3 (already in simplest form) Denominator: 2y^2+y-1 = (2y-1)(y+1) Step 2: Rewrite the expression with the factored terms. ((y+3)(y-2))/((y-2)(y+2)) + (y+3)/((2y-1)(y+1)) Step 3: Simplify the first fraction by canceling the common factor (y-2). (y+3)/(y+2) + (y+3)/((2y-1)(y+1)) Step 4: Find a common denominator to add the fractions. The common denominator is (y+2)(2y-1)(y+1). ((y+3)(2y-1)(y+1))/((y+2)(2y-1)(y+1)) + ((y+3)(y+2))/((y+2)(2y-1)(y+1)) Step 5: Combine the numerators over the common denominator. ((y+3)(2y-1)(y+1) + (y+3)(y+2))/((y+2)(2y-1)(y+1)) Step 6: Factor out the common term (y+3) from the numerator. ((y+3)[(2y-1)(y+1) + (y+2)])/((y+2)(2y-1)(y+1)) Step 7: Expand and simplify the expression inside the square brackets in the numerator. (2y-1)(y+1) + (y+2) = (2y^2 + 2y - y - 1) + (y+2) = (2y^2 + y - 1) + (y+2) = 2y^2 + 2y + 1 Step 8: Substitute this back into the expression. ((y+3)(2y^2+2y+1))/((y+2)(2y-1)(y+1)) The final answer is ((y+3)(2y^2+2y+1))/((y+2)(2y-1)(y+1)). Part (ii): What must be added to x^2+6x to make it a perfect square? Step 1: Recall the formula for a perfect square trinomial. A perfect square trinomial is of the form (x+a)^2 = x^2 + 2ax + a^2. Step 2: Compare the given expression x^2+6x with the first two terms of the perfect square form. We have x^2+6x. Comparing this to x^2+2ax, we can equate the coefficients of x: 2a = 6 Step 3: Solve for a. a = (6)/(2) a = 3 Step 4: Determine the term a^2 that completes the square. a^2 = 3^2 a^2 = 9 Step 5: The value that must be added to x^2+6x to make it a perfect square is 9. This results in x^2+6x+9, which is (x+3)^2. The final answer is 9.