With the simplified equation, use appropriate techniques to solve it. (Since this particular equation will have an term on the left side, this is a quadratic equation. So we would use techniques for solving quadratic equations:
Get one side of the equation equal to zero.
Factor the other side (or use the Quadratic Formula).
If you factored in step 2, then use the Zero Product Property to finish the solution.
But there is a faster way with this equation which I will show you first.
(3r + 9)(r - 1) = -(21r - 21)
First we will get one side equal to zero by adding (21r-21) to each side:
(3r + 9)(r - 1) + (21r - 21) = 0
Next I will factor out 21 from (21r - 21):
(3r + 9)(r - 1) + 21(r - 1) = 0
Note how there is a factor of (r-1) in each of the two parts of the left side of the equation. (It is because I noticed this common factor that I saw this shortcut.) We can factor out (r-1):
(r - 1)*((3r + 9) + 21) = 0
The second factor simplifies to:
(r - 1)*(3r + 30) = 0
We can now use the Zero Product Property which tells us that this (or any) product can be zero only if one (or more) of the factors is zero. So:
r-1 = 0 or 3r+30 = 0
Solving these we get:
r = 1 or r = -10
If (r-1) had not been a common factor (or if we did not notice that it was) then we would have to solve the equation using the procedure I outlined above:
(3r + 9)(r - 1) = -(21r - 21)
1) Simplify
2) Since this is a quadratic equation we want one side to be zero. Adding 21r and subtracting 21 we get:
This will factor. First the Greatest Common Factor (GCF), which is 3:
Then the trinomial factors:
3(r-1)(r+10) = 0
Using the Zero Product Property:
3 = 0 or r-1 = 0 or r+10 = 0
There is no solution to the first equation. But we do get solutions from the other two:
r = 1 or r = -10
(Note that these are the same as the answers we got from the faster method.)
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