Factoring ax^2 + bx + c
Find two different values that complete each expression so that the trinomial can be factored into the product of two binomials. Factor your trimonials.
4n^2+ __n - 3
Please explain and solve.
I multiplied the first and last to get -12
(4)(-3) = -12
now what factors give me -12 ?
could be 4, -3 or -4, 3 or 6, -2 or -6, 2
for each pair, their sum has to go in the middle, so
we could have:
4n^2 + n - 3
4n^2 -n - 4
4n^2 + 4n - 3
4n^2 - 4n - 3
each of these WILL factor
e.g.
4n^2 + 4n - 3 = (2n-1)(2n+3)
To factor the trinomial 4n^2 + __n - 3, we need to find two numbers that, when multiplied, give us -12 (the product of the constants) and when added, give us the coefficient of the middle term (-1n).
We can start by listing all the possible pairs of factors of -12:
(1, -12), (-1, 12), (2, -6), (-2, 6), (3, -4), (-3, 4)
Next, we need to determine which pair of factors satisfies the condition of having a sum of -1. By inspecting the list, we can see that the pair (-3, 4) satisfies this condition because -3 + 4 = 1.
Now we can rewrite the middle term -1n as the sum of -3n and 4n:
4n^2 - 3n + 4n - 3
Next, we group the terms and factor by grouping:
(4n^2 - 3n) + (4n - 3)
n(4n - 3) + 1(4n - 3)
Notice that we have a common binomial factor of (4n - 3) in both terms. We can factor this out:
(4n - 3)(n + 1)
So, the fully factored form of the trinomial 4n^2 - 1n - 3 is (4n - 3)(n + 1).
To solve the trinomial, we set each factor equal to zero and solve for n:
Setting (4n - 3) = 0, we get n = 3/4.
Setting (n + 1) = 0, we get n = -1.
Therefore, the trinomial is factored as (4n - 3)(n + 1), and the values that complete each expression are n = 3/4 and n = -1.