factor completely
3x^2-14x-24
I do not see the gcf for the problem
I am totally lost can some explain please I have several these problems and i need to figure out how to solve them
You put
3x^2-14x-24 = (a x + b)(c x + d)
Equating the coefficients on both ides gives:
3 = a c
This means that we can take
a = 3 and c = 1:
3x^2-14x-24 = (3 x + b)(x + d)
Check that you don't get anything new if you take a = -3 and c = -1 or if you interchange a and c.
Equating the coefficient of x on both sides gives:
3 d + b = -14
Equating the constant term on both sides gives:
b d = -24
Some trial and error yields:
b=4 and d = -6 d, so we obtain:
3x^2-14x-24 = (3 x + 4)(x -6)
When factoring using this method you use trial and error. The factorization of the constant term, in this case -24 plays an important role. What you are using is that since the factorization has to be true for all x, it has to be true also for x = 0.
This means that you can often simply matters considerably by considering some other point instead of x = 0. In this case, if you take x = -1 then the function is -7 which has a unique factorization (up to signs)
So, let's see how that works out in this case. Let's start here:
3x^2-14x-24 = (3 x + b)(x + d)
If you substitute x = 0 in both sides then you get the equation:
-24 = b d
But there are many possibilities here. So, let's take x = -1 instead. We then get:
-7 = (b - 3)(d - 1)
Possibility 1:
b-3 = -7 and d-1 = 1
doesn't work because then b = -4 and d = 2 and b d isn't equal to 24.
Possibility 2:
b-3 = 7 and d-1 = -1
a non-starter because then d becomes zero and the product b d = 0.
Possibility 3:
b-3 = 1 and d-1 = -7
This is a possible solution because you get b = 4 and d = 6 and b d = 24.
You still need to check that
3 d + b = -14
when solving the problem in this way.
Typo:
You put
3x^2-14x-24 = (a x + b)(c x + d)
Equating the coefficients of x^2 on both sides gives:
3 = a c
A few more typos in the last part of the solution:
Possibility 1:
b-3 = -7 and d-1 = 1
doesn't work because then b = -4 and d = 2 and b d isn't equal to -24.
Possibility 2:
b-3 = 7 and d-1 = -1
a non-starter because then d becomes zero and the product b d = 0.
Possibility 3:
b-3 = 1 and d-1 = -7
This is a possible solution because you get b = 4 and d = -6 and b d = -24.
To factor the quadratic expression 3x^2-14x-24 completely, follow these steps:
Step 1: Identify the GCF (Greatest Common Factor) of the three terms.
In this case, there is no common factor among the coefficients 3, -14, and -24. So, we need to proceed with a different method.
Step 2: Use the ac-method or trial and error to find the product of two numbers that multiply to give ac (product of the coefficient of x^2 and the constant term) and add up to b (the coefficient of x). In this case, ac is (-24)(3) = -72 and b is -14.
Step 3: Look for two numbers whose product is -72 and sum is -14 by trial and error. In this case, the numbers are -4 and 18 because (-4)(18) = -72 and -4 + 18 = 14.
Step 4: Rewrite the middle term (-14x) using the two numbers found in step 3.
Replace -14x with -4x + 18x in the original expression:
3x^2 - 4x + 18x - 24
Step 5: Group the terms and factor by grouping.
Factor out the GCF from the first two terms and the last two terms:
x(3x - 4) + 6(3x - 4)
Step 6: Notice that the terms in parentheses, (3x - 4), are the same in both groups. Factor out the common binomial:
(3x - 4)(x + 6)
Therefore, the completely factored form of 3x^2 - 14x - 24 is (3x - 4)(x + 6).