Factor:
20+22v-12v^2
-12V^2 + 22V + 20.
Factor using AC Method:
2(-6V^2 + 11V + 10),
A * C = -60 = -4 * 15,
2(-6V^2 + (-4V + 15V) + 10,
Group the 4 terms into 2 factorable
pairs:
2((-6V^2 + 15V) + (-4V + 10))
2(3V(-2V + 5) + 2(-2V + 5)),
2(-2V + 5) (3V + 2)
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To factor the expression 20 + 22v - 12v^2, we can follow the steps below:
Step 1: Rearrange the terms in descending order of powers.
Start by arranging the terms from highest to lowest powers of v:
-12v^2 + 22v + 20.
Step 2: Look for a common factor in all three terms.
In this case, there is no common factor among -12v^2, 22v, and 20.
Step 3: Check if the quadratic expression can be factored further.
To determine if the quadratic expression can be factored further, we need to find the factors of the coefficient of v^2 (-12) and the constant term (20).
To find the factors of -12, we consider the pairs of numbers whose product is -12. The possible pairs are:
-1 and 12,
1 and -12,
-2 and 6,
2 and -6,
-3 and 4,
3 and -4.
Since the middle term of the quadratic expression is positive, one of the factors must be positive, and the other must be negative.
Next, we examine the pairs of numbers above and their sums to find which pair adds up to the coefficient of v (22). The pair that works in this case is 2 and -6 since 2 + (-6) = -4.
Step 4: Split the middle term using the selected pair and factor by grouping.
Using the pair 2 and -6, we rewrite the middle term (22v) as:
22v = 2v - 6v.
Now we can rewrite the expression and group the terms:
-12v^2 + 2v - 6v + 20.
Step 5: Factor by grouping and factor out common terms.
Now we group the terms together and factor out a common factor from each group:
-2v(6v - 1) - 2(3v - 10).
Step 6: Factor out the common factor.
Factor out the common factor (-2):
-2(6v - 1) - 2(3v - 10).
The factored form of the expression 20 + 22v - 12v^2 is:
-2(6v - 1) - 2(3v - 10).