Factor: 12a^2-17a-5
the factorization will be one of
(12a- )(a+ )
(12a+ )(a- )
(6a- )(2a+ )
(6a+ )(2a- )
(3a- )(4a+ )
(3a+ )(4a- )
where the missing numbers are 1 and 5
a little trial produces
(3a-5)(4a+1)
This will be factored into an expression of the form
(A*a + B)(C*a + D)
Multiplying out:
A*C*a^2 + A*D*a + B*C*a + B*D =
A*C*a^2 + (A*D + B*C)*a + B*D =
12a^2-17a-5
which means
A*C = 12
A*D + B*C = -17
B*D =-5
Then start trying whole numbers. . .B and D have to be +/- 1 or +/-5
A and C have to be positive common factors of 12: so 1 and 12 or 3 and 4 or 6 and 2. . .
-4*5 + 3*1 = -17. . .
A = 4, B = 1, C = 3, D = -5
The factored expression is
(4a + 1)*(3a - 5)
To factor the expression 12a^2 - 17a - 5, we need to find two binomials that can be multiplied together to give us the given expression.
Step 1: Multiply the coefficient of the leading term (12) by the constant term (-5). In this case, we have 12 * -5 = -60.
Step 2: Find two numbers that multiply to give us -60 and add up to the coefficient of the middle term (-17). In this case, the numbers are -20 and 3, because -20 * 3 = -60 and -20 + 3 = -17.
Step 3: Rewrite the middle term (-17a) using the numbers found in step 2. We split the middle term into -20a + 3a.
Now, we can rewrite the expression as follows:
12a^2 - 20a + 3a - 5
Step 4: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair.
(12a^2 - 20a) + (3a - 5)
Step 5: Factor out the GCF from each pair.
4a(3a - 5) + 1(3a - 5)
Now notice that we have a common binomial factor, (3a - 5).
Step 6: Combine the two terms with the common binomial factor.
(4a + 1)(3a - 5)
Therefore, the factored form of 12a^2 - 17a - 5 is (4a + 1)(3a - 5).