SOLUTION: how would i find which one is the factor of the trinominal? 6x^2+19x-36. Your help will be greatly appreciated. Thank you.

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Question 1160031: how would i find which one is the factor of the trinominal?
6x^2+19x-36. Your help will be greatly appreciated. Thank you.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52802) About Me  (Show Source):
You can put this solution on YOUR website!
.

Solve the equation

    6x^2 + 19x - 36 = 0.


It is not so simple to factorize it --- therefore, use the quadratic formula.


In this way, you will find two roots  x%5B1%5D = 4%2F3  and  x%5B2%5D = -4.5.


Therefore, the factorization is


    6x^2 + 19x - 36 = 6%2A%28x-4%2F3%29%2A%28x-%28-4.5%29%29 = 6%2A%28x-4%2F3%29%2A%28x%2B4.5%29 = (3x-4)*(2x+9).


It is your factorization, and the factors are  (3x-4)  and  (2x+9).    ANSWER

Solved.

---------------

On quadratic formula for solving quadratic equations see the lessons
    - Introduction into Quadratic Equations
    - PROOF of quadratic formula by completing the square
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic  "Quadratic equations".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Your statement of the problem suggests that answer choices are given. If so, re-post your question, showing those choices.

The process of finding the factorization of a quadratic is different if answer choices are given than it is if they are not.

If answer choices are given, then you can look at them and quickly determine which one(s) will work. If choices are not given, then finding the factorization will (for most students) be more work.

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Many students will go straight to the quadratic formula to factor a quadratic expression like this, as tutor @ikleyn did in her response.

I prefer to spend a little time playing with possible factorizations to find the right one.

This example is greatly complicated by the fact that both the leading coefficient 6 and the constant term 36 have many different pairs of factors.

However, most of the possible linear factors based solely on the factors of 6 and 36 can be eliminated by logical reasoning.

For example, consider the possible linear factors of the form 6x+a, where a is a factor of 36:
6x+1
6x+2
6x+3
6x+4
6x+6
6x+9
6x+12
6x+18
6x+36

In every one of those possible linear factors except the first, there is a common factor. But the given quadratic has no common factor among all three coefficients, so those are not possible factors of the quadratic.

So the only POSSIBLE factor containing "6x" is 6x+1; or it might be 6x-1.

But we can quickly determine that neither of those will work, knowing that the constant term of the given quadratic is -36:
(6x+1)(x-36) = 6x^2-215x-36
(6x-1)(x+36) = 6x^2+215x-36

Neither of those products has the correct linear term. It is helpful, however, to note that switching the signs in the two factors switches the sign of the linear term in the product.

So now let's look at linear factors of the form 2x+a:
2x+1
2x+2
2x+3
2x+4
2x+6
2x+9
2x+12
2x+18
2x+36

Again ruling out any linear factor containing a common factor, the only possibilities are 2x+1, 2x+3, and 2x+9 (or they might be 2x-1, 2x-3, or 2x-9).

Again we try each of these, knowing that the constant term in the quadratic is -36:

(2x+1)(3x-36)... Oops! We don't need to try that one, because 3x-36 has a common factor
(2x+3)(3x-12)... Oops again, for the same reason...
(2x+9)(3x-4) = 6x^2+19x-36 AHA! That's it!!

All the words of explanation make this look like a long and tedious process; but with a little practice, some good mental math skills, and logical reasoning to discard possible linear factors that contain a common factor, it can go fast.