What Multiplies to But Adds To: The Factoring Trick Every Algebra Student Needs
If you have ever stared at a problem asking, “Find two numbers that multiply to 12 but add to 7,” you already know how confusing it can feel at first. Also, the phrase “what multiplies to but adds to” is one of the most searched math questions online, and for good reason — it is the hidden engine behind factoring quadratics, simplifying expressions, and solving equations that show up in almost every algebra class. In simple terms, the question is asking you to find a pair of numbers that satisfy two conditions at the same time: their product (the result of multiplication) equals one value, and their sum (the result of addition) equals another. Once you learn the pattern, what looks like guesswork becomes a fast, reliable skill.
This is the bit that actually matters in practice.
What Does “What Multiplies to But Adds To” Actually Mean?
The idea is easier than it sounds. You are given two clues:
- Clue 1 (Multiplication): Two numbers, when multiplied together, give a specific product.
- Clue 2 (Addition): The same two numbers, when added together, give a specific sum.
To give you an idea, if someone asks, “What two numbers multiply to 20 but add to 9?Consider this: the answer is 4 and 5, because 4 × 5 = 20 and 4 + 5 = 9. This leads to ” you are looking for a pair that fits both rules. The challenge grows when negative numbers enter the picture or when the product is large, but the logic stays exactly the same.
Why This Skill Matters in Algebra
This “multiply and add” thinking is not just a random puzzle — it is the foundation of factoring quadratic expressions. In real terms, a quadratic like x² + 7x + 12 is factored by finding two numbers that multiply to 12 (the constant term) and add to 7 (the coefficient of the middle term). Those numbers, 3 and 4, become the values inside the parentheses: (x + 3)(x + 4).
Real talk — this step gets skipped all the time.
Without this skill, students are forced to rely on slower methods such as completing the square or the quadratic formula for every problem. Mastering the multiply-add connection makes factoring faster, builds number sense, and strengthens mental math confidence.
Step-by-Step Method to Find the Two Numbers
Follow this simple process every time you face a “what multiplies to but adds to” problem.
Step 1 — Identify the Product and the Sum
Write down the two target values clearly. To give you an idea, if the question says, “multiply to 18, add to 9,” your product is 18 and your sum is 9.
Step 2 — List All Factor Pairs of the Product
Factor pairs are two whole numbers that multiply together to produce the product. For 18, the pairs are:
- 1 and 18
- 2 and 9
- 3 and 6
Step 3 — Check Which Pair Adds to the Target Sum
Now add each pair:
- 1 + 18 = 19
- 2 + 9 = 11
- 3 + 6 = 9 ✅
The pair 3 and 6 matches both conditions No workaround needed..
Step 4 — Verify Your Answer
Always double-check by multiplying and adding again. 3 × 6 = 18 and 3 + 6 = 9. Both clues are satisfied, so the answer is correct.
Understanding the Sign Rules
The trickiest part of “what multiplies to but adds to” problems is handling negative numbers. Three scenarios cover almost every case.
Both Numbers Are Positive
When the product is positive and the sum is positive, both numbers are positive. Example: multiply to 15, add to 8 → 3 and 5.
Both Numbers Are Negative
When the product is positive but the sum is negative, both numbers are negative. Example: multiply to 12, add to –7 → –3 and –4, because (–3) × (–4) = 12 and (–3) + (–4) = –7 That alone is useful..
One Positive, One Negative
When the product is negative, the two numbers must have opposite signs. That's why the sign of the sum tells you which number has the larger absolute value. Example: multiply to –10, add to 3 → 5 and –2, because 5 × (–2) = –10 and 5 + (–2) = 3.
Worked Examples You Can Try
Example 1: What multiplies to 24 but adds to 10? Factor pairs of 24: (1, 24), (2, 12), (3, 8), (4, 6). The pair 4 and 6 adds to 10, so the answer is 4 and 6 Not complicated — just consistent..
Example 2: What multiplies to –18 but adds to 3? Since the product is negative, one number is positive and one is negative. Try pairs of 18: (1, 18), (2, 9), (3, 6). The combination 6 and –3 gives 6 × (–3) = –18 and 6 + (–3) = 3. Answer: 6 and –3.
Example 3: What multiplies to 36 but adds to –12? Product positive, sum negative → both numbers negative. Pairs of 36 include (–
6, 6), (–6, –6), (–4, –9), (–3, –12), (–2, –18), (–1, –36). The pair –6 and –6 adds to –12, so the answer is –6 and –6.
Putting It All Together
Now let’s combine everything into a single, more complex problem.
Example 4: What multiplies to –28 but adds to –3?
- Product = –28 → one positive, one negative.
- Sum = –3 → the negative number has the larger absolute value.
- List factor pairs of 28: (1, 28), (2, 14), (4, 7).
- Try combinations: 4 and –7 give 4 × (–7) = –28 and 4 + (–7) = –3.
Answer: 4 and –7.
Why This Skill Matters
This “multiply-add” technique isn’t just a standalone trick—it’s the foundation for factoring quadratic expressions like (x^2 + bx + c). Also, in that context, you’re looking for two numbers that multiply to (c) and add to (b). By mastering the simpler numerical version first, you build the mental agility needed for algebra.
It also sharpens your number sense. The more you practice spotting factor pairs and checking their sums, the faster you’ll recognize relationships between numbers—a skill useful in everything from mental arithmetic to advanced problem-solving.
Quick Practice Problems
Test your new skills with these:
- Multiply to 30, add to 11 → 5 and 6
- Multiply to –24, add to 5 → 8 and –3
- Multiply to 49, add to –14 → –7 and –7
- Multiply to –32, add to –4 → 4 and –8
Final Thoughts
The “what multiplies to but adds to” puzzle is more than a math exercise—it’s a gateway to algebraic fluency. By breaking the problem into clear steps—identifying the product and sum, listing factor pairs, checking the sum, and applying sign rules—you can solve even tricky cases with confidence. Which means with practice, this process becomes second nature, turning what once felt like guesswork into a reliable, logical method. Keep practicing these problems, and you’ll find that factoring quadratics and simplifying expressions in algebra become far less intimidating But it adds up..
