factor trinomial completely of the type ax^2+bx+c.

-9+18x^2+21x

I am not able to solve the solution since the steps are quite confusing any help is great

first arrange them in the standard order

18x^2 + 21x - 9

look for a common factor, in this case 3

18x^2 + 21x - 9
= 3(6x^2 + 7x - 3)

A method which seems to be popular these days is "decomposition", (not my personal favourite)

Step1: multiply the (a)(c)
(6)(-3) = -18
step2: look for 2 factors which multiply to get -15, and whose sum is the middle term coefficent
-18 = 6(-3), 6-3 = 3 , no
= 2(-9) , 2 - 9 = -7 , no
= -2(9) , -2+9 = 7 , yeahhh the sum is 7

If you cannot find such a combination, the quadratic does not factor

step3: split the middle term into the sum of these two
3(6x^2 - 2x + 9x - 3)
= 3( 2x(3x - 1) + 3(3x - 1) )
notice that 3x-1 is a common factor, this MUST happen, if not, you made a mistake
= 3(3x-1)(2x+3)

small typo

should have said:

step2: look for 2 factors which multiply to get -18, and whose sum is the middle term coefficient

But you probably figured that out anyway.

Thanks for your help!

I understand it much better.

To factor a trinomial of the type ax^2+bx+c, we need to find two binomials that, when multiplied together, give us the original trinomial. In this case, the trinomial is -9+18x^2+21x.

Step 1: Make sure the trinomial is written in standard form, which means arranging the terms in descending order of their exponents. In this case, the trinomial is already in standard form, so no rearrangement is necessary.

Step 2: Look for a common factor, if any, among all the terms. In this case, there is no common factor among the terms.

Step 3: We need to find two numbers whose product is equal to the product of the coefficient of x^2 (18) and the constant term (-9). In this case, the product is -9 * 18 = -162.

Step 4: Next, we want to find two numbers whose sum is equal to the coefficient of x (21). In this case, the sum is 21.

Step 5: We need to find the factors of -162 that add up to 21. These factors are 27 and -6.

Step 6: Rewriting the middle term (21x) using the factors found in the previous step, we get:
18x^2 + 27x - 6x - 9

Step 7: Now we group the terms in pairs and factor out the greatest common factor (GCF) from each pair:
(18x^2 + 27x) + (-6x - 9)

Step 8: Factoring out the GCF from each pair, we end up with:
9x(2x + 3) - 3(2x + 3)

Step 9: Notice that we now have a common binomial factor, (2x + 3).

Final Step: We can factor out this common binomial factor from both terms and obtain the fully factored form:
(2x + 3)(9x - 3)

Therefore, the given trinomial -9+18x^2+21x factors as (2x + 3)(9x - 3).