Welcome! Today we're learning The Splitting-the-Middle-Term Procedure — the workhorse method for factoring quadratics.

You know that factoring a polynomial and setting each factor to zero gives you the zeros.

But how do you factor in the first place?

The standard technique is splitting the middle term:

Step	What you do
1	Replace $bx$ with two terms
2	Their coefficients multiply to $a \times c$
3	Their coefficients add to $b$
4	Grouping does the rest

This is the workhorse method — it handles every factorable quadratic you will encounter in this chapter.

1. Computing a*c and finding the pair

Let's check your understanding of the splitting the middle term method for factoring quadratics.

The method begins with two key computations:

Calculate the product $a \times c$
Find two numbers whose product equals $a \times c$ AND whose sum equals $b$

This is the critical decision point of the entire factoring process.

📋 Given Info

Here's an example to remind you how it works:

For $2 x^{2} + 5 x - 12$ :

$a = 2$ , $b = 5$ , $c = - 12$
Compute $a \times c = 2 \times (-12) = -24$
Find two numbers with product $-24$ and sum $5$

Since the product is negative, one number must be positive and the other negative.

The pair is $8$ and $-3$ found it!
- Check: $8 \times (- 3) = - 24$ ✓
- Check: $8 + (- 3) = 5$ ✓

✍️ Question

Your turn! 🧮

For $3 x^{2} + 10 x + 8$ :

Compute $a \times c$
Find two numbers whose product equals $a \times c$ AND whose sum equals $b$
Verify both conditions

Show your work!

Here is the systematic approach to finding the pair:

Step 1: Compute $a \times c = 3 \times 8 = 24$ .

Step 2: Note the target sumThe coefficient b is your target sum: $b = 10$ targetThis is what your two numbers must add to.

✍️ MCQ

Choose one

We need two numbers with product

24

and sum

10

. Since the product is positive and the sum is positive, what can we conclude about the signs of both numbers?

✓

Step 3: Since $a \times c = 24 > 0$ The sign of the product tells you what to look for and $b = 10 > 0$ The sign of b guides your search, both numbers must be positiveBoth positive means both factors are positive.

Step 4: List factor pairs of 24Two numbers that multiply to 24: $(1, 24)$ , $(2, 12)$ , $(3, 8)$ , $(4, 6)$ .

Step 5: Check sums: $1 + 24 = 25$ , $2 + 12 = 14$ , $3 + 8 = 11$ , $4+6 = 10$ This pair sums to our target. Found it!We found the matching pair

The pair is 6 and 4answerThe two numbers we need: product $= 24$ ✓Check that product equals 24, sum $= 10$ ✓Check that sum equals 10.

✍️ MCQ

Choose one

For

2x^2 + 5x - 12

, we need two numbers with product

-24

and sum

5

. Which pair works?

✓

Sign rule:Tells you what kind of numbers to look for

If $a \times c > 0$ and $b > 0$ Check these signs first to narrow your search, both numbers are positiveBoth factors will be positive.
If $a \times c > 0$ and $b < 0$ Product positive but sum negative, both numbers are negativeBoth factors will be negative.
If $a \times c < 0$ Negative product means mixed signs, one number is positive and one is negativeYou need one of each sign.

✍️ MCQ

Choose one

For

2x^2 + 5x - 12

, we have

a \times c = -24

(negative). According to the sign rule, what can we say about the two numbers we need?

✓

2. Splitting, grouping, and extracting the common factor

Once you have the pair, the rest is mechanical:

Split the middle term
Group into two pairs
Extract the GCF from each pair
Extract the common binomial factor

This is the second half of the procedure.

📋 Given Info

For $3 x^{2} + 10 x + 8$ , the pair is 6 and 4.

Split: $3x^2 + 6x + 4x + 8$

Now it's your turn to group and extract common factors.

✍️ Question

Complete the factoring:

Split $3x^2 + 10x + 8$ using the pair $(6, 4)$ , then group and extract the common factor.

Show your work step by step.

After finding the pair $(6, 4)$ , the grouping is mechanicalNo thinking required, just follow the steps:

SplitRewrite as two separate terms that add up to the original: $3 x^{2} + 10 x + 8 = 3 x^{2} + 6 x + 4 x + 8$

$10x = 6x + 4x$ The sum must equal the original coefficient

(sum=10) $6 + 4 = 10$ That's how you verify you did it right

✍️ MCQ

Choose one

Why did we split

10x

into

6x + 4x

specifically, and not some other combination like

5x + 5x

?

✓

GroupWe group to create pairs that share a common factor: $(3 x^{2} + 6 x) + (4 x + 8)$

Extract GCF from each group:

From $3 x^{2} + 6 x$ : GCF is $3 x$ . Factor: $3 x (x + 2)$
From $4 x + 8$ : GCF is $4$ . Factor: $4 (x + 2)$

(Both groups must have the same bracket after factoring)

✍️ MCQ

Choose one

After extracting the GCF from both groups, we got

3x(x + 2) + 4(x + 2)

. What is the common binomial factor?

✓

Both groups share the common factor $(x + 2)$ This shared factor confirms our splitting worked. Extract it:

$3 x (x + 2) + 4 (x + 2) =$ $<pen actions="circle" circle-color="green" circle-annotation="final" narrationText="x plus 2 times 3 x plus 4" commentary="Product of two simpler expressions">(x + 2)(3x + 4)$ (Two linear factors multiplied together) $< / p e n >$

✍️ MCQ

Choose one

What is the fully factored form of

3x^2 + 10x + 8

?

✓

If the two groups had different binomials — say $3 x (x + 2)$ and $4 (x + 1)$ — that would mean the pair $(6, 4)$ was wrong. Go back and try another pair. The grouping step is a built-in error check.If binomials don't match, you picked the wrong pair

3. Handling negative a*c (one positive, one negative number)

Now let's tackle a slightly trickier case!

When $a \times c$ is negative, the pair of numbers we're looking for must have one positive and one negative number.

This is a bit harder because you need to get both the product AND the sum right with mixed signstricky!.

📋 Given Info

Here's your challenge:

For the quadratic $2 x^{2} - x - 6$ :

$a = 2$ , $b = - 1$ , $c = - 6$
$a \times c = 2 \times (-6) = -12$

Since $a \times c < 0$ , one number must be positive and the other negative.

✍️ Question

Factor $2 x^{2} - x - 6$ by splitting the middle term.

Show:

The pair of numbers (product $= -12$ , sum $= -1$ )
The split middle term
The final factored form

When $a \times c$ is negativeA negative product is your signal, the two numbers you're looking for must have opposite signsWhat kind of factor pairs to look for — one positive and one negative.

Why opposite signs?

For the product to be negative, one factor must be positive and one must be negativeOnly way to get a negative product.

In our example: $a \times c = 2 \times (-6) = -12$ Don't check pairs with same signs

Since $- 12 < 0$ , we need numbers like $(+ 3)$ and $(- 4)$ where:

Product: $(+ 3) \times (- 4) = - 12$ ✓
Sum: $(+ 3) + (- 4) = - 1 = b$ ✓

✍️ MCQ

Choose one

For

2x^2 - x - 6

, we found the pair

(+3, -4)

. What is the correct way to split the middle term

-x

?

✓

For $2 x^{2} - x - 6$ : we have $a \times c = 2 \times (-6) = -12$ a times c gives you the target product, and $b = -1$ b is your target sum.

The factor pairs of 12 are: $(1, 12)$ , $(2, 6)$ , $(3, 4)$ .

✍️ MCQ

Choose one

We need two numbers with product

-12

and sum

-1

. Since the sum is negative and small, which number should be negative — the larger one or the smaller one?

✓

Since the product must be negative, one number is positive and one is negative. We need the sum to be $- 1$ , so the larger numberlarger factor gets the negative sign (in absolute value) should be negativelarger factor gets the negative sign.

This means we're looking at: $+ 1$ and $- 12$ ? Sum is $- 11$ . Too negative. $+ 2$ and $- 6$ ? Sum is $- 4$ . Still too negative. $+ 3$ and $- 4$ ? Sum is $- 1$ . That's it!

Let's verify our pair: $- 4$ and $+ 3$ .

Product: $(- 4) \times 3 = - 12$ ✓matches!Must verify the product matches
Sum: $- 4 + 3 = - 1$ (exact!)✓(Must verify the sum matches too)

Both conditions are satisfied!If either fails, wrong pair

🎯 These are our numbers!These numbers are what we use next

The pair $(- 4, + 3)$ will split the middle termBreak one term into two for grouping $- x$ into $-4x + 3x$ So you can group and factor.

Now we complete the factoring:

SplitThis is the crucial step in the method: $2 x^{2} -$ $4x + 3x$ (These multiply to negative 12 and add to negative 1) $- 6$

Group(Just pair terms together in sequence): $($ $2x^2 - 4x$ First pair stays together $) + ($ $3x - 6$ Second pair stays together $)$

ExtractPull out the greatest common factor: $2x$ GCF(Both groups must yield matching brackets) $(x - 2) +$ (GCF) $3$ (Check your split if brackets don't match) $(x - 2)$

✍️ MCQ

Choose one

Both groups have the common factor

(x - 2)

. When we factor this out from

2x(x - 2) + 3(x - 2)

, what will be the other factor?

✓

Common factorThe shared factor that makes the method work: $(x - 2)(2x + 3)$ The completely factored expression

The zerosWhat we solve for at the end are $x =$ $2$ zeroSet x minus 2 equals zero and $x =$ $-\frac{3}{2}$ zeroSet 2x plus 3 equals zero.