Math Calculators, Lessons and Formulas

It is time to solve your math problem

mathportal.org

Trinomials Factoring Calculator

google play badge app store badge

This calculator factors trinomials of the form $ax^2+bx+c$ using the AC method and a formula $ax^2+bx+c = a(x-x_1)(x-x_2)$, where $x_1$ and $x_2$ are solutions of a quadratic equation. Calculator shows all the work and provides detailed explanation for each step.

Factor trinomial $$ \color{blue}{ 3x^2-5x+2 } $$

solution

The factored form is $$ \color{blue}{ 3x^2-5x+2 = \left(x-1\right)\left(3x-2\right) } $$

explanation

Step 1: Identify constants $ a $ , $ b $ and $ c $. ( $ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant). In our case:

$$ a = 3 , b = -5 ~ \text{ and } ~ c = 2 $$

Step 2: Multiply the constant term by the leading coefficient.

$$ a \cdot c = 6 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 6 $ and add to $ b = -5 $.

Step 4: All pairs of numbers with a product of $ 6 $ are:

PRODUCT = 6
1     6 -1     -6
2     3 -2     -3

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -5 }$

PRODUCT = 6 and SUM = -5
1     6 -1     -6
2     3 -2     -3

Step 6: Replace middle term $ -5 x $ with $ -2x-3x $:

$$ 3x^{2}-5x+2 = 3x^{2}-2x-3x+2 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first group and $ -1 $ out of the second group.

$$ 3x^{2}-2x-3x+2 = x\left(3x-2\right) -1\left(3x-2\right) = \left(x-1\right) \left(3x-2\right) $$

Report an Error !

Script name : factoring-trinomials-calculator

Form values: 3 , 2 , 5 , 1 , 2 , g , Factor trinomial 3x^2-5x+2 , Factor 3x^2-5x+2

Comment (optional)

Share Result

You can copy and paste the link wherever you need it.

Quadratic trinomials factoring
factors trinomials of the form $ax^2 + bx + x$.
help ↓↓ examples ↓↓ tutorial ↓↓
4x2 - 20x + 25
x2 - 8x + 15
2x2-11x+12

New service on Mathportal

You can hire our experts to do your math homework.

We provide complete, handwritten, step-by-step solutions.

Get a free quote
working...
EXAMPLES
example 2:ex 2:
Factor $16x^2 + 16x + 1$
example 2:ex 2:
Write trinomial $2 x^2 - 5x - 3 $ in factored form.
example 2:ex 2:
Factor $6 x^2 +13x - 5 $
Search our database of more than 200 calculators
TUTORIAL

Polynomial Factoring Techniques

This calculator factors trinomials of the form $ ax^2 + bx + c $ using the methods listed below.

1. Factoring perfect square trinomial

2. Factor if leading coefficient $ a = 1 $

3. Factor if leading coefficient $ a \ne 1 $

4. Special cases ( $ b = 0 $ ) or ( $ a = 0 $ )

Method 1 : Factoring perfect square trinomial

Example 01: Factor $ 4a^2 - 12a + 9 $

Step1: Verify that both the first and third terms are perfect squares.

$4a^2$ is perfect square because $4a^2 = \left(\color{blue}{2a}\right)^2$

$9$ is perfect square because $9 = \left(\color{red}{3}\right)^2 $

Step2: Check if middle term is twice the product of $ \color{blue}{2a} $ and $ \color{red}{3} $

$$ \text{middle term} = 12a = 2 \cdot \color{blue}{2a} \cdot \color{red}{3} $$

Step3: Put $ \color{blue}{2a} $ and $ \color{red}{3}$ inside parentheses. Because the middle term's coefficient is negative, we'll insert a minus sign inside parenthesis.

$$ 4a^2 - 12a + 9 = ( \color{blue}{2a} - \color{red}{3} )^2 $$

Method 2 : Leading coefficient $ a = 1 $

In this case, the trinomial has the following form $ x^2 + bx + c $.

Example 02: Factor $ x^2 + 7x + 10 $

To factor this trinomial we need to find two integers ( $p$ and $q$ ) such that $ p + q = b $ and $ p \cdot q = c $.

In this example $ p + q = 7 $ and $p \cdot q = 10$

After some trials and errors we get $ p = 2 $ and $ q = 5 $

The factored form is

$$ x^2 + 7x + 10 = ( x + p)(x + q) = (x + 2)(x + 5) $$

Method 4 : Special Cases

Example 04: Factor $ 3x^2 + 5x $

This is special case where $c = 0$.

To solve this one we just need to factor $x$ out of $ 3x^2 + 5x $

$$ 3x^2 + 5x = x ( 3x + 5) $$

Example 05: Factor $ 25x^2 - 4 $

This is special case where $b = 0$.

We'll need to use the difference of squares formula to factor this one.

$$ 25x^2 - 4 = (5x)^2 - 2^2 = (5x-2)(5x+2) $$

Method 3 : Leading coefficient $ a \ne 1 $

In this case, the trinomial has the following form: $ ax^2 + bx + c $.

Example 03: Factor $ 3x^2 - 5x + 2 $

Step 1: Identify constants $a$ , $ b $ and $c$

$$ a = 3, b = -5 , c = 2 $$

Step 2: Find out two numbers ( $p$ and $q$) that multiply to $ a \cdot c = 6 $ and add up to $ b = -5 $.

After some trials and errors we get $ \color{blue}{p = -2} $ and $ \color{red}{q =-3} $

Step 3:Replace middle term ( $ -5x $ ) with $ \color{blue}{-2}x \color{red}{-3}x $

$$ 3x^2 - 5x + 2 = 3x^2 - 2x - 3x + 2 $$

Step 4:Factor out x from the first two terms and -1 from the last two terms.

$$ \begin{aligned} 3x^2 - 5x + 2 &= 3x^2 - 2x - 3x + 2 = \\\\ &= x(3x-2) -1(3x-2) = \\\\ &=(x - 1)(3x-2) \end{aligned} $$
Search our database of more than 200 calculators

Was this calculator helpful?

Yes No
438 172 776 solved problems