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Factoring Quadratic Expressions

Factoring quadratic expressions is one of the most fundamental "power skills" you need for the Digital SAT. While it might seem like a basic algebra topic

Tutor AI Team2026-02-23T05:22:58.683Z

Factoring quadratic expressions is one of the most fundamental "power skills" you need for the Digital SAT. While it might seem like a basic algebra topic you learned in 9th grade, the SAT elevates this concept by placing it within the Advanced Math domain under Equivalent Expressions. On a typical Digital SAT, you can expect to see 2 to 4 questions that directly require factoring, and several more where factoring is a necessary step to solve a larger problem, such as finding the zeros of a function or simplifying a rational expression.

In the context of the Digital SAT, "factoring" isn't just about getting an answer; it’s about recognition. The test-makers want to see if you can look at a complex-looking expression like $6x^2 - 11x - 10$ and realize it is actually the same thing as $(3x + 2)(2x - 5)$ . This skill is vital because the SAT often asks you to identify "equivalent expressions" or to find the "zeros" of a quadratic function. If you can't factor, you are forced to use the Quadratic Formula, which is time-consuming and prone to calculation errors.

In this guide, we are going to move beyond the simple $x^2 + bx + c$ trinomials. We will focus on the "Medium" and "Hard" level variations: trinomials where the leading coefficient is not 1 ( $ax^2 + bx + c$ ) and the method of Factoring by Grouping. By the end of this guide, you will have a systematic approach to breaking down any quadratic the SAT throws at you, whether you are working by hand or using the built-in Desmos calculator. We will transform factoring from a "guess and check" chore into a precise, repeatable process.

Core Concepts

To master factoring on the SAT, you must be comfortable with three primary techniques. None of these formulas are provided on the SAT Reference Sheet, so you must commit the methods to memory.

1. Factoring Trinomials where $a = 1$

Before tackling harder problems, you must be fluent in the basic form: $x^2 + bx + c$ To factor this, you look for two numbers that multiply to $c$ and add to $b$ .

SAT Recognition: This appears when the question asks for the $x$ -intercepts or "zeros" of a parabola.
Example: $x^2 - 5x + 6$ . Factors of $6$ that add to $-5$ are $-2$ and $-3$ . Thus, $(x - 2)(x - 3)$ .

2. Factoring Trinomials where $a \neq 1$ (The "AC Method")

This is the "Medium" difficulty level that frequently appears on the SAT. When the expression looks like $ax^2 + bx + c$ , you cannot simply look at the last number. You must use the AC Method:

Multiply the leading coefficient $a$ and the constant $c$ (the product $ac$ ).
Find two numbers that multiply to $ac$ and add to $b$ .
Split the middle term $bx$ into two terms using these numbers.
Factor by grouping (see below).

Example: $2x^2 + 7x + 3$

$a \cdot c = 2 \cdot 3 = 6$ .
Find factors of $6$ that add to $7$ : $6$ and $1$ .
Rewrite: $2x^2 + 6x + 1x + 3$ .
Group: $(2x^2 + 6x) + (1x + 3) \Rightarrow 2x(x + 3) + 1(x + 3) \Rightarrow (2x + 1)(x + 3)$ .

3. Factoring by Grouping

This technique is essential for four-term polynomials and is the second half of the AC Method.

When to use: When you see four terms, or after you have split the middle term of a trinomial.
The Process:
1. Group the first two terms and the last two terms.
2. Factor out the Greatest Common Factor (GCF) from each pair.
3. If the binomial inside the parentheses matches, factor that binomial out.

Formulaic View: $ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)$

4. Difference of Squares and Perfect Square Trinomials

The SAT loves "Special Products" because they allow students who recognize patterns to save time.

Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$ $a^{2} - b^{2} = (a - b) (a + b)$
- Recognition: Two terms, both perfect squares, separated by a minus sign.
- Example: $9x^2 - 16 = (3x - 4)(3x + 4)$ .
Perfect Square Trinomials:
- $a^2 + 2ab + b^2 = (a + b)^2$
- $a^2 - 2ab + b^2 = (a - b)^2$
- Recognition: The first and last terms are perfect squares, and the middle term is twice the product of their square roots.

5. The Greatest Common Factor (GCF) First

The most common mistake students make is forgetting to check for a GCF before starting.

Rule: Always look for a number or variable that goes into ALL terms.
Example: $4x^2 - 20x + 24$ . Factor out the $4$ first: $4(x^2 - 5x + 6)$ , then factor the inside to $4(x - 2)(x - 3)$ .

SAT Strategy Tips

1. The "Desmos Shortcut" for Zeros

On the Digital SAT, you have access to the Desmos graphing calculator for the entire math section. If a question asks for the factors of $ax^2 + bx + c$ , you can graph the expression.

The $x$ -intercepts (roots) of the graph relate directly to the factors.
If the graph crosses the $x$ -axis at $x = k$ , then $(x - k)$ is a factor.
Warning: If the intercept is a fraction like $x = 0.5$ , the factor is $(2x - 1)$ , not $(x - 0.5)$ . To find this, set $x = \frac{1}{2}$ , multiply by $2$ to get $2x = 1$ , then subtract $1$ to get $2x - 1 = 0$ .

2. Working Backward from Answer Choices

Since the SAT is largely multiple-choice, you can often "FOIL" or multiply the answer choices to see which one matches the original expression. This is a "fail-safe" strategy if you forget the AC Method. However, only use this if you are fast at mental math, as it can be a time-sink.

3. Look at the Constants

If you are factoring $ax^2 + bx + c$ into $(mx + p)(nx + q)$ , remember that $p \cdot q$ must equal $c$ . You can often eliminate 2 out of 4 answer choices just by looking at the last numbers in the binomials.

4. The "Structure" of the Question

The SAT often uses "structure" questions. For example: "If $12x^2 + ax - 20 = (3x + 4)(4x - 5)$ , what is the value of $a$ ?" Instead of factoring the left side (which has an unknown), simply expand the right side using FOIL: $(3x + 4)(4x - 5) = 12x^2 - 15x + 16x - 20 = 12x^2 + 1x - 20$ By comparison, $a = 1$ .

Worked Example: Medium): Typical Trinomial Factoring

Problem

Question: Which of the following is an equivalent form of the expression $3x^2 - 10x - 8$ ? A) $(3x - 2)(x + 4)$ B) $(3x + 2)(x - 4)$ C) $(3x - 4)(x + 2)$ D) $(3x + 4)(x - 2)$

Solution

Identify $a, b,$ and $c$ : $a = 3, b = -10, c = -8$ .
Find $ac$ : $3 \cdot (-8) = -24$ .
Find factors of $-24$ that add to $-10$ :
- $-12$ and $2$ work because $-12 \cdot 2 = -24$ and $-12 + 2 = -10$ .
Split the middle term: $3x^2 - 12x + 2x - 8$ .
Factor by grouping:
- Group 1: $3x^2 - 12x \Rightarrow 3x(x - 4)$
- Group 2: $2x - 8 \Rightarrow 2(x - 4)$
Combine: $(3x + 2)(x - 4)$ .
Match with options: The correct answer is B.

Worked Example: Hard): GCF and Multi-Step Factoring

Problem

Question: If the expression $10x^2 + 55x - 105$ is rewritten in the form $k(ax + b)(cx + d)$ , where $k, a, b, c,$ and $d$ are integers, what is the value of $a + c$ ?

Solution

Factor out the GCF: All terms are divisible by $5$ . $5(2x^2 + 11x - 21)$
Factor the inner trinomial ( $2x^2 + 11x - 21$ ):
- $ac = 2 \cdot (-21) = -42$ .
- Find factors of $-42$ that add to $11$ : $14$ and $-3$ .
Split and Group: $2x^2 + 14x - 3x - 21$ $2x(x + 7) - 3(x + 7)$ $(2x - 3)(x + 7)$
Reassemble the full expression: $5(2x - 3)(x + 7)$
Identify $a$ and $c$ : The coefficients of $x$ are $2$ and $1$ .
Calculate $a + c$ : $2 + 1 = 3$ . Final Answer: 3

Worked Example: SAT-Hard): Factoring by Grouping with Variables

Problem

Question: Which of the following is equivalent to the expression $x^3 - 4x^2 - 9x + 36$ ? A) $(x - 4)(x - 3)(x + 3)$ B) $(x + 4)(x - 3)(x + 3)$ C) $(x - 4)(x^2 + 9)$ D) $(x - 6)(x + 6)(x - 1)$

Solution

Recognize the four-term structure: This is a prime candidate for factoring by grouping.
Group the first two and last two terms: $(x^3 - 4x^2) + (-9x + 36)$
Factor out the GCF from each group:
- From $(x^3 - 4x^2)$ , factor out $x^2$ : $x^2(x - 4)$ .
- From $(-9x + 36)$ , factor out $-9$ : $-9(x - 4)$ . (Note: We factor out a negative to make the parentheses match).
Combine the groups: $(x^2 - 9)(x - 4)$
Look for further factoring: Notice that $(x^2 - 9)$ is a Difference of Squares. $(x^2 - 9) = (x - 3)(x + 3)$
Final factored form: $(x - 3)(x + 3)(x - 4)$
Match with options: The correct answer is A.

Practice Problems

Which of the following is equivalent to $4x^2 - 13x + 3$ ? A) $(4x - 1)(x - 3)$ B) $(4x - 3)(x - 1)$ C) $(2x - 3)(2x - 1)$ D) $(4x + 1)(x + 3)$
If $6x^2 + 7x - 20 = (ax + b)(cx + d)$ for all values of $x$ , where $a, b, c,$ and $d$ are constants, what is the value of $ad + bc$ ? (Note: This is a "Grid-In" style question)
The expression $2x^3 + 3x^2 - 8x - 12$ can be written as $(2x + 3)(x + k)(x - k)$ , where $k$ is a positive constant. What is the value of $k$ ? A) 2 B) 3 C) 4 D) 6

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Common Mistakes

1. Forgetting the GCF

Students often jump straight into the AC Method without checking if they can simplify the numbers first.

Wrong: Factoring $10x^2 - 40$ using complex methods.
Right: Factoring out $10$ first: $10(x^2 - 4) = 10(x - 2)(x + 2)$ .
Why it matters: Smaller numbers reduce the chance of arithmetic errors.

2. Sign Errors in Grouping

When factoring by grouping, if the third term is negative, you must factor out a negative GCF.

Example: $x^2(x - 5) - 4x + 20$ .
Mistake: $x^2(x - 5) + 4(x + 5)$ . (The signs don't match!)
Correction: $x^2(x - 5) - 4(x - 5) = (x^2 - 4)(x - 5)$ .

3. Confusing Factors and Roots

The SAT will often ask for the "sum of the solutions" or "zeros."

Mistake: Factoring $x^2 - 5x + 6$ into $(x - 2)(x - 3)$ and saying the solutions are $-2$ and $-3$ .
Correction: If the factor is $(x - 2)$ , the solution is $x = 2$ . Always flip the sign!

4. Incorrect AC Method Split

Students sometimes find factors that multiply to $ac$ but don't add to $b$ .

Example: For $2x^2 + 5x - 3$ , $ac = -6$ .
Mistake: Using $2$ and $3$ (they add to $5$ but multiply to $+6$ , not $-6$ ).
Correction: Use $6$ and $-1$ (they multiply to $-6$ and add to $5$ ).

Frequently Asked Questions

Can I just use the Quadratic Formula for every factoring question?

Technically, yes, the Quadratic Formula ( $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ ) will give you the roots. If the roots are $x_1$ and $x_2$ , the factors are $(x - x_1)$ and $(x - x_2)$ . However, this is often much slower than factoring by hand, especially on the Digital SAT where time is of the essence. Use the formula as a backup, not a primary tool.

How does factoring connect to the graph of a parabola?

Factoring is the bridge between the "Standard Form" ( $ax^2 + bx + c$ ) and the "Intercept Form" ( $a(x - p)(x - q)$ ). The factors tell you exactly where the graph crosses the $x$ -axis. If you see a graph on the SAT and need to find its equation, look at the $x$ -intercepts and turn them into factors.

What if a quadratic can't be factored?

On the SAT, if a question asks you to find "equivalent expressions" or "factors," the expression will almost certainly be factorable using integers. If the question asks for "solutions" and the answer choices contain square roots, that is your signal that the expression is not factorable and you should use the Quadratic Formula or Desmos.

Key Takeaways

✓
Always check for a GCF first. It’s the easiest way to turn a "Hard" problem into a "Medium" one.
✓
Master the AC Method: Multiply $a \cdot c$ , find factors that add to $b$ , split the middle, and group.
✓
Difference of Squares is everywhere. Memorize $a^2 - b^2 = (a - b)(a + b)$ . It is the SAT's favorite shortcut.
✓
Use Desmos strategically. Graph the expression to find $x$ -intercepts if you get stuck factoring by hand.
✓
Factors vs. Zeros: If $(x - k)$ is a factor, then $x = k$ is a zero. Don't mix up the signs.
✓
Check your work with FOIL. If you have time, multiply your factors back together to ensure they match the original expression.

Tags:

#sat#math#advanced-math

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Core Concepts

1. Factoring Trinomials where a=1a = 1a=1

2. Factoring Trinomials where a≠1a \neq 1a=1 (The "AC Method")

3. Factoring by Grouping

4. Difference of Squares and Perfect Square Trinomials

5. The Greatest Common Factor (GCF) First

SAT Strategy Tips

1. The "Desmos Shortcut" for Zeros

2. Working Backward from Answer Choices

3. Look at the Constants

4. The "Structure" of the Question

Worked Example: Medium): Typical Trinomial Factoring

Worked Example: Hard): GCF and Multi-Step Factoring

Worked Example: SAT-Hard): Factoring by Grouping with Variables

Practice Problems

Common Mistakes

1. Forgetting the GCF

2. Sign Errors in Grouping

3. Confusing Factors and Roots

4. Incorrect AC Method Split

Frequently Asked Questions

Key Takeaways

Related Topics

Transition Words and Phrases

Analyzing Author's Purpose

Vocabulary in Context

Ready to Ace Your SAT math?

1. Factoring Trinomials where $a = 1$

2. Factoring Trinomials where $a \neq 1$ (The "AC Method")