Question

Solve the following equation by grouping

`x^{2}+5x+6=0`

Answer 1

To solve the equation `x^{2} + 5x + 6 = 0` by grouping, we need to find two numbers that multiply to `6` and add up to `5` (since the coefficient of the middle term is `5` and the constant term is `6`). The numbers that satisfy these conditions are `2` and `3`.


We can then rewrite the middle term by splitting it into two terms using these two numbers:


`x^{2} + 2x + 3x + 6 = 0`


Now we can group the terms:


`(x^{2} + 2x) + (3x + 6) = 0`


Next, we factor out the greatest common factor from each group:


`x(x + 2) + 3(x + 2) = 0`


Notice that the terms `x + 2` are the same in both groups. We can factor this out as well:


`(x + 2)(x + 3) = 0`


Now, we set each factor equal to zero and solve for `x`:


`x + 2 = 0` or `x + 3 = 0`


For the first equation, subtracting `2` from both sides gives:


`x = -2`


For the second equation, subtracting `3` from both sides gives:


`x = -3`


Therefore, the solutions to the equation `x^{2} + 5x + 6 = 0` are `x = -2` and `x = -3`.