I need to know how to factor these in A-C method form
2a^2+1+3a
9w-w^3
I've never encountered this AC method before, but I've tried to google it. It's easy though it's crazy. XD
Anyway,
#1.
I'll change the variable into x so it won't be confusing:
2x^2 + 3x + 1
First step is to multiply a (the numerical coeff of x^2) by c (the constant). Then change a into 1. Here, a is equal to 2 and the c is 1:
x^2 + 3x + 1(2)
x^2 + 3x + 2
Second step is to factor this new expression:
(x+1)(x+2)
Third step is to change the numerical coeff of x by the original a (which is 2):
(2x+1)(2x+2)
Fourth step & final step is to further factor the expression above, and divide by the original a:
(2x+1)(2x+2) / 2
2(2x+1)(x+1) / 2
(2x+1)(x+1)
And it's actually correct (when you factor it normally).
Now try solving #2. We can rewrite the equation as
-w(w^2 - 9)
You'll use the AC method only in the w^2 - 9. Here, a = 1 and c = -9.
Hope this helps :3
2a^2+1+3a
A = 2
B = 3
C = 1
A* c
2* 1 = 2
1* 2
-1 -2
Sum
3
-3
B = 3
1 * 2
(a + 1) (2a+ 1)
To factor expressions in the A-C method form, you need to look for a quadratic expression with two terms (a binomial) in the given expression.
Let's start with the expression: 2a^2 + 1 + 3a.
Step 1: Group the terms.
Rearrange the terms to group the quadratic term together and the constant term separately:
2a^2 + 3a + 1.
Step 2: Split the middle term.
Multiply the coefficient of the quadratic term (2) with the constant term (1) to get (2)(1) = 2.
Now, find two numbers that multiply to 2 and add up to the coefficient of the linear term (3). In this case, the numbers are 2 and 1 because 2 * 1 = 2 and 2 + 1 = 3.
Step 3: Rewrite the expression.
Replace the middle term (3a) with the numbers found in the previous step (2a + 1a):
2a^2 + 2a + 1a + 1.
Step 4: Group the terms again.
Now, group the first two terms and the last two terms:
(2a^2 + 2a) + (1a + 1).
Step 5: Factor out the common factors.
Factor out the greatest common factor from each group:
2a(a + 1) + 1(a + 1).
Step 6: Combine the terms.
Since both groups now have a common factor of (a + 1), combine the two groups into one expression:
(2a + 1)(a + 1).
Therefore, the factored form of the expression 2a^2 + 1 + 3a using the A-C method is (2a + 1)(a + 1).
Let's move on to the second expression: 9w - w^3.
Step 1: Reorganize the terms.
Rearrange the terms so that the highest-degree term comes first:
-w^3 + 9w.
Step 2: Factor out the common factor.
In this case, the common factor is -w, so we can factor it out:
-w(w^2 - 9).
Step 3: Apply the difference of squares.
The expression (w^2 - 9) can be further factored as a difference of squares:
-w(w - 3)(w + 3).
Therefore, the factored form of the expression 9w - w^3 using the A-C method is -w(w - 3)(w + 3).
Remember, factoring requires practice. The more you work with different expressions, the better you'll become at recognizing patterns and applying the appropriate factoring techniques.