What Multiplies to 24 and Adds to 10: The Fast Way to Solve It
You're doing your homework, and there it is — that classic algebra problem. In practice, find two numbers that multiply to 24 but add up to 10. Your first instinct might be to guess randomly. 12 and 2? They multiply to 24, sure, but add to 14. In real terms, too high. Practically speaking, 6 and 4? That's 6×4=24, and 6+4=10. Got it Turns out it matters..
Short version: it depends. Long version — keep reading.
But what if the problem was "multiplies to 24 and adds to 14"? Or "multiplies to 24 and adds to -2"? That's where things get interesting — and where most students get stuck But it adds up..
Here's the thing: once you understand the pattern behind these problems, you can solve any version instantly. No more checking every possibility. Also, no more guessing. Just a quick method that works every time.
What Does "Multiplies to 24 and Adds to" Actually Mean?
This is really a factoring problem in disguise. When you see "find two numbers that multiply to 24 and add to [x]," you're being asked to do one thing: factor a quadratic equation That alone is useful..
The standard form looks like this:
x² + (sum)x + (product) = 0
So when someone says "multiplies to 24 and adds to 10," they're describing a quadratic where:
- The constant term (the number at the end) is 24
- The coefficient of x (the middle number) is 10
The equation would be x² + 10x + 24 = 0, and you're looking for two numbers that replace the x terms — numbers that multiply to 24 and add to 10.
This comes up constantly in algebra: factoring trinomials, solving quadratic equations, simplifying expressions. It's one of those foundational skills that makes everything else click.
Why This Problem Shows Up Everywhere
Here's why you keep seeing this type of question. When you factor x² + 10x + 24, you're looking for:
(x + a)(x + b) = x² + (a+b)x + ab
So a and b need to multiply to 24 and add to 10. That's exactly the problem.
Once you find those two numbers — 6 and 4 — you can factor the whole expression:
x² + 10x + 24 = (x + 6)(x + 4)
That's the answer. That's what the problem is really asking you to do.
How to Solve It: The Method That Actually Works
Let's walk through the process step by step, then I'll show you the shortcut that makes it instant The details matter here..
Step 1: List the Factor Pairs of 24
Start by writing out every pair of integers that multiply to 24:
- 1 and 24
- 2 and 12
- 3 and 8
- 4 and 6
- 6 and 4 (same as above, just reversed)
- 8 and 3
- 12 and 2
- 24 and 1
If you're dealing with negative numbers, you'd also include pairs like -1 and -24, -2 and -12, and so on.
Step 2: Add Each Pair
Now add them up:
- 1 + 24 = 25
- 2 + 12 = 14
- 3 + 8 = 11
- 4 + 6 = 10 ← This is our answer
- 6 + 4 = 10
- 8 + 3 = 11
- 12 + 2 = 14
- 24 + 1 = 25
The pair that adds to your target sum — in this case, 10 — is 4 and 6.
That's the method. It works every time.
The Faster Way: Mental Shortcut
Once you've done this a few times, you'll notice a pattern. You're looking for two numbers close to each other that multiply to your target. Why? Because when numbers are closer together, their sum is smaller.
Think about it: 4 and 6 are pretty close, and their sum is 10. Compare that to 1 and 24, which are far apart — their sum is 25, much larger.
So when you're trying to find numbers that multiply to 24 and add to a small sum (like 10 or 11), look for the factor pairs that are closest together. That usually gets you there faster Small thing, real impact. Took long enough..
Other Versions You'll Encounter
The problem doesn't always ask for "adds to 10." Here are the most common variations:
Multiplies to 24, adds to 14
Using the same method:
- 1 + 24 = 25
- 2 + 12 = 14 ← Got it
- 3 + 8 = 11
- 4 + 6 = 10
The answer is 2 and 12.
So x² + 14x + 24 = (x + 2)(x + 12).
Multiplies to 24, adds to 11
Check the pairs:
- 3 + 8 = 11 ← This works
The answer is 3 and 8 Which is the point..
So x² + 11x + 24 = (x + 3)(x + 8) It's one of those things that adds up..
Multiplies to 24, adds to -10
At its core, where negative numbers come in. When the sum is negative but the product is positive, both numbers must be negative:
- -1 + (-24) = -25
- -2 + (-12) = -14
- -3 + (-8) = -11
- -4 + (-6) = -10 ← This works
The answer is -4 and -6.
So x² - 10x + 24 = (x - 4)(x - 6).
Multiplies to 24, adds to -2
When the product is positive (24) but the sum is negative, you need both numbers negative. But -4 + -6 = -10, not -2. And -12 + -2 = -14. Let me check negative factor pairs again...
Actually, there's no integer pair that multiplies to 24 and adds to -2. The closest you can get is -4 + -6 = -10. This is one of those problems with no solution using whole numbers — you'd need to move into fractions or decimals, which is a different kind of problem That's the whole idea..
That's worth knowing: not every combination has an answer. Some sums simply don't work with integer factors.
Common Mistakes Students Make
Here's where most people go wrong:
1. Forgetting to check both positive and negative versions
If your sum is negative but your product is positive, you need two negative numbers. Students often forget this and only check positive pairs, then get frustrated when nothing works.
2. Stopping after finding one pair
With 24, you have several factor pairs. Don't assume the first one you find is right — check that it actually adds up to your target number.
3. Not considering that the order doesn't matter
6 and 4 gives you the same result as 4 and 6. When you're making your list, you can skip the reversed duplicates to save time That's the part that actually makes a difference..
4. Confusing the sum with the product
It sounds obvious, but under pressure, students sometimes multiply when they should add, or vice versa. Double-check which one you're solving for It's one of those things that adds up..
Practical Tips for Solving These Fast
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Memorize the factor pairs of common numbers — 24, 36, 48, 60, 72. These come up constantly in algebra. Knowing them instantly saves you time on every problem.
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When the sum is small, look for factors close together — 4 and 6, 3 and 8. When the sum is large, look for factors far apart — 1 and 24, 2 and 12 Simple, but easy to overlook. Simple as that..
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Check the sign of your target sum first — If it's negative and your product is positive, you know both factors are negative. If your product is negative, one factor is positive and one is negative Worth knowing..
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Practice with different numbers — Once you master 24, try the same method with 18, 36, or 48. The process is identical; you're just working with different factor pairs.
FAQ
What two numbers multiply to 24 and add to 10?
The answer is 6 and 4. They multiply to 24 (6×4=24) and add to 10 (6+4=10) Simple, but easy to overlook..
What multiplies to 24 and adds to 14?
The answer is 12 and 2. They multiply to 24 (12×2=24) and add to 14 (12+2=14).
What multiplies to 24 and adds to 11?
The answer is 8 and 3. They multiply to 24 (8×3=24) and add to 11 (8+3=11) Simple, but easy to overlook. Took long enough..
What multiplies to 24 and adds to -10?
The answer is -6 and -4. They multiply to 24 [(-6)×(-4)=24] and add to -10 (-6+-4=-10).
How do I factor x² + 10x + 24?
Find two numbers that multiply to 24 and add to 10 — that's 6 and 4. Then write: (x + 6)(x + 4).
The next time you see one of these problems, you'll know exactly what to do. List your factor pairs, check the sums, and pick the one that matches. It only takes a few seconds once you get the hang of it — and now you've got the pattern down Practical, not theoretical..
Not obvious, but once you see it — you'll see it everywhere.
