Factoring Trinomial Equations Calculator

What is a Trinomial Equation?

A trinomial equation is like a math puzzle with three pieces! It's an equation that has three terms, usually in the form of \(ax^2 + bx + c = 0\). Think of it as a special type of math sentence where we need to find the value of x that makes the sentence true.

How to Factor a Trinomial Equation

Factoring a trinomial is like breaking a big number into smaller parts that multiply together. We're looking for two expressions that, when multiplied, give us our original trinomial. It's like finding two smaller puzzles that fit together to make our big puzzle!

Formula

A general trinomial equation looks like this:

\[ ax^2 + bx + c = 0 \]

Where:

• \(a\) is the coefficient of \(x^2\) (the number in front of \(x^2\))
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term (the number without an \(x\))
• \(x\) is the variable we're trying to find

Calculation Steps

1. Look at the signs of \(b\) and \(c\)
2. Find two numbers that multiply to give \(ac\) and add to give \(b\)
3. Rewrite the middle term using these two numbers
4. Factor by grouping
5. Check your answer by multiplying the factors

Example and Visual Representation

Let's factor this trinomial: \(x^2 + 5x + 6\)

We need to find two numbers that multiply to give 6 and add to give 5.

These numbers are 2 and 3.

So, we can rewrite our equation as: \(x^2 + 2x + 3x + 6\)

Now, let's group these terms: \((x^2 + 2x) + (3x + 6)\)

Factor out the common terms: \(x(x + 2) + 3(x + 2)\)

Our final factored form is: \((x + 2)(x + 3)\)

Let's show this with a picture:

In this picture, we can see how \((x + 2)\) and \((x + 3)\) multiply to give us our original trinomial \(x^2 + 5x + 6\). The area of the whole rectangle represents our trinomial, while the sides represent the factors we found!