using factorization method: solve b²+5b+6=0 and show specific workings

Sure, here is the step-by-step solution for the quadratic equation b²+5b+6=0 using the factorization method:

Step 1: Write down the equation b² + 5b + 6 = 0

Step 2: Identify the coefficients of b², b, and the constant term. In this case, they are 1, 5, and 6 respectively.

Step 3: Find two numbers that multiply to give the constant term (6) and add up to give the coefficient of b (5). The numbers that satisfy these conditions are 2 and 3 because 2*3 = 6 and 2+3 = 5.

Step 4: Rewrite the middle term of the equation (5b) as the sum of the terms 2b and 3b. This gives us: b² + 2b + 3b + 6 = 0

Step 5: Factor by grouping. The first two terms can be grouped together, and the last two terms can be grouped together. This gives us: b(b + 2) + 3(b + 2) = 0

Step 6: Notice that (b + 2) is a common factor. Factor out (b + 2) to get: (b + 2)(b + 3) = 0

Step 7: Set each factor equal to zero and solve for b. This gives us: b + 2 = 0 or b + 3 = 0

Solving these equations gives us: b = -2 or b = -3

So, the solutions to the equation b² + 5b + 6 = 0 are b = -2 and b = -3.