Question

How do you factor the trinomial `10x^2+11x-6=0`?

Answer 1

`(5x-2)(2x+3)=0`

Explanation:

We can factor by grouping. To do this in a scenario like this, we must look for two numbers that meet the following characteristics:

• Have a product the same as the product of the first and last coefficients `(10xx-6=mathbf(-60))`
• Have a sum the same as the middle term `(mathbf11)`

Examine all the factor pairs to determine which factors of `-60` these could be. Remember, since the product of the numbers is negative, one of the numbers must be negative too. That means the sum of `11` will be created by a difference of the numbers.

Possible factor pairs:

`{:(-1 and 60,",",1 and -60),(-2 and 30,",",2 and -30),(-3 and 20,",",3 and -20),(color(red)(-4 and 15),",",4 and -15),(-5 and 12,",",5 and -12),(-6 and 10,",",6 and -10):}`

The pair `-4,15` is the only whose sum is `-11`.

Since `15x-4x=11x`, we can replace `11x` in the original trinomial.

`10x^2+color(blue)(11x)-6=0`

`10x^2+color(blue)(15x-4x)-6=0`

Now, we can factor by grouping. Sort the trinomial into to groups of two.

`(10x^2+15x)-(4x+6)=0`

Notice the change in sign on the constant `-6`, since we factored a negative.

Now, factor a common term from each set in the parentheses.

`5x(2x+3)-2(2x+3)=0`

Factor out a common `(2x+3)` term. In a sense, this "combines" the `5x` and `-2` terms.

`color(green)((5x-2)(2x+3))=0`

This is the trinomial, completely factored. This step could be used to solve the equation.