Factoring a Quadratic Trinomial

Factoring a Quadratic Trinomial

How to factor a quadratic trinomial: examples and their solutions.

Example 1

Factor the given polynomial. x^2 + 3x + 2

Write (x)(x),
leaving some space after each x.

Think of a pair of numbers
whose product is the constant term: +2.
+1 and +2 seems to be good.
So write + 1 and + 2 in each space.

The x term found by the FOIL method is,
+2x + x, +3x.

See if this is equal to
the given trinomial's middle term: +3x.

It is equal to +3x.

So (x + 1)(x + 2) is the answer.

Finding the right pair of numbers is the triky part.
Here's one of the hints you can use:

Hint #1:
If the middle term and the last term are both (+),
then the pair of numbers are both (+).

Example 2

Factor the given polynomial. y^2 - 5y + 6

Hint #2:
If the middle term is (-) and the last term is (+),
then the pair of numbers are both (-).

Write (y)(y),
leaving some space after each y.

Think of a pair of numbers
whose product is the constant term: +6.
-1 and -6 seems to be good.
So write - 1 and - 6 in each space.

The y term found by the FOIL method is,
-6y - y, -7y.

This is not equal to
the given trinomial's middle term: -5y.

Factor the given polynomial. y^2 - 5y + 6

Then find another pair of numbers
whose product is +6.
-2 and -3 seems to be good.
So write - 2 and - 3 in each space.

The y term found by the FOIL method is,
-3y - 2y, -5y.

This is equal to
the given trinomial's middle term: -5y.

So (y - 2)(y - 3) is the answer.

Just like this example,
these problems are solved by trial and error.

Example 3

Factor the given polynomial. a^2 - a - 12

Hint #3:
If the last term is (-),
then the pair of numbers are (+) and (-).

Write (a)(a),
leaving some space after each a.

Think of a pair of numbers
whose product is the constant term: -12.
-4 and -3 seems to be good.
So write - 4 and - 3 in each space.

The a term found by the FOIL method is,
-4a + 3a, -a.

This is equal to
the given trinomial's middle term: -a.

So (a - 4)(a + 3) is the answer.

Example 4

Factor the given polynomial. 2x^2 + 7x + 6

Write (2x)(x),
leaving some space after 2x and x.

Think of a pair of numbers
whose product is the constant term: +6.
+3 and +2 seems to be good.
So write + 3 and + 2 in each space.

The x term found by the FOIL method is,
+4x + 3x, +7x.

This is equal to
the given trinomial's middle term: +7x.

So (2x + 3)(x + 2) is the answer.

Example 5

Factor the given polynomial. x^2 + 5xy - 24y^2

Write (xy)(xy),
leaving some space between each x and y.

Think of a pair of numbers
whose product is the constant term: +24.
+8 and -3 seems to be good.
So write + 8 and - 3 in each space.

The xy term found by the FOIL method is,
-3xy + 8xy, +5xy.

This is equal to
the given trinomial's middle term: +5xy.

So (x + 8y)(x - 3y) is the answer.