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how do you solve a trinomial equation?

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3 Answers

When factoring trinomials that are in the form of ax2+bx+c=0 these are the steps you will need to take:

1. See if the coeffiecients a, b, and c all have any common factors that you can factor out.

For example, the coeffiecients of 2x2+12x+6=0 all have a common factor of 2, so you can factor the 2 out to get:

2(x2+6x+3)=0.

In the case of 8x2+2x-3=0, we cannot factor anything out since 8, 2, and -3 have no common factors.

2. Multiply a*c and list out all of the factor pairs of the product.

In this case a=8 and c=-3, so a*c=-24. The factor pairs of 24 are: -1 and 24, 1 and -24, -2 and 12, 2 and -12, -3 and 8, 3 and -8, -4 and 6, and 4 and -6.

3. Find which pair of factors' sum is equal to b.

In this case b=2. So we want the pair of factors' sum to equal (positive) +2. If we choose the pair -4 and 6, and add them to eachother, we get -4+6= 2. So this is the pair of factors we will need to use.

4. In the original equation, write bx in terms of the sums of the factors you chose.

In this case 2x=-4x+6x. So we will write the original equation as:

8x2-4x+6x-3=0.

5. Group the first two terms together in a set of paranthesis, and the second two terms together in a set of separate paranthesis and factor out the greatest common factors for each set of paranthesis.

(8x2-4x)+(6x-3)=0

4x(2x-1)+3(2x-1)=0     Whatever is left inside of the paranthesis should be equal to eachother, so if your paranthesis do not match you know that you did something wrong. In this case we have 2x-1 in both of our paranthesis so we know we are on the right track.

6. Add the terms that you factored out of the paranthesis and multiply by the paranthesis that has a match.

So we have (4x+3)(2x-1)=0. This is the factored form of your trinomial.

7. To solve the trinomial, set each paranthesis equal to 0 and solve for x.

So,

4x+3=0

4x=-3

x=-3/4

and

2x-1=0

2x=1

x=1/2

So your final answer is x=-3/4 and x=1/2.

 

 

Just in case you don't get the "binomials equal to 0 part"  this fact that you can use the two factors to solve for x comes from the "zero property of multiplication".

If two numbers (known as factors) give a product of zero when multiplied together, then the only way that could happen is if one, or both, or those numbers is, itself, equal to zero.

Knowing this allows you take one factor at a time and pretend that it is your star candidate for finding a solution.  You set it equal to zero, and then you solve.  

Lots of students get frustrated late at night, and scream out, "WHY DO I HAVE TO LEARN TO FACTOR, ANYWAY!?!"   The answer is because factoring creates linear factors of x, equations with only one solution.  

Because you can set these factors equal to zero, they give unique solutions, one at a time, even when multiple solutions exist.

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