B^2 + 35 = −12b Solve each equation by factoring I really need help with understanding this step by step, thank you. I am giving 40 points for this.

Step-by-step explanation:

We have the equation and we want to solve it by factoring.

First, let's move all the stuff to one side so that the other side is 0. So, let's add to both sides. This yields:

Let's review how to factor. If we have an equation in the following form:

Where a, b, and c are the coefficients of the variable (in our example, our variable is b), then we must find two numbers, p and q, such that:

Our equation can be rewritten as:

So, our a is 1, b is 12, and c is 35.

Therefore, we need to find two numbers that when multiplied together yields a(c) = 1(35) = 35 and we added together yields b = 12.

From here, we just have to guess and check. We can start by listing all the factors of 35. There aren't in fact that many:

1 + 35 is 36, not 12. However, 5 + 7 is indeed 12. So, our two numbers p and q are 5 and 7.

Now what we've found our two numbers, we substitute the b term for our two numbers. We have:

We will substitute 12b for 5b + 7b:

From here, we factor by grouping. From the first two terms, factor out a b:

And from the last two terms, factor out a 7:

Now, notice that both terms have a . So, by using grouping or the reverse of the distribute property, we can write:

Notice that if we distribute the left term into the right, we get , so our equation is indeed equivalent.

So now we. have:

We can now use the Zero Product Property to acquire:

Solve for b for each case. Therefore, the solutions of our equation are:

Notes:

If we can't find two numbers p and q that satisfy our conditions, this means that the equation cannot be factored. So, we will use alternative methods to solve our equation.