The negative square root of 400 is -20 Small thing, real impact..
That's the answer. You can close this tab now if that's all you needed Nothing fancy..
But here's the thing — most people who search this aren't just looking for a number. Even so, they're studying for a test, helping a kid with homework, or trying to remember why there are two square roots for every positive number. And somewhere along the way, the distinction between "the square root" and "a square root" got fuzzy.
So let's talk about it. Properly Not complicated — just consistent..
What Is a Square Root, Really?
Before we dig into negative roots, let's make sure we're on the same page about what a square root is.
A square root of a number is a value that, when multiplied by itself, gives you the original number. In practice, that's it. No fancy language needed Simple, but easy to overlook..
So for 400, we're asking: what number times itself equals 400?
20 × 20 = 400.
(-20) × (-20) = 400 That's the part that actually makes a difference..
Both work. Both are square roots of 400.
The Principal Square Root vs. The Negative Square Root
Here's where textbooks and teachers start using specific language — and where students start getting confused Nothing fancy..
The principal square root is the non-negative one. By convention, when you see the radical symbol √400, that symbol only refers to the positive root. So √400 = 20. Always. Never -20.
The negative square root is the other one. It's written as -√400 = -20.
Notice the placement of that minus sign. And √400 is 20. It matters. -√400 is -20. They are not the same thing.
And if someone asks "what are the square roots of 400?And " — plural — the answer is both: 20 and -20. Or written compactly: ±20.
Why Does This Convention Exist?
It's not arbitrary. It's about functions.
A function, by definition, gives you exactly one output for each input. If √x gave you two numbers, it wouldn't be a function. Calculus, algebra, computer programming — they all rely on √x being a single, predictable value.
So mathematicians agreed: the radical symbol means "the principal (non-negative) square root." Full stop.
If you want the negative one, you write the minus sign outside the radical. -√x.
Simple? Yes.
Easy to forget under pressure? Also yes Not complicated — just consistent..
Why the Negative Square Root Matters
You might wonder: if the principal root is the "official" one, why do we even care about the negative version?
Solving Equations
This is the big one.
Say you're solving x² = 400.
If you take the square root of both sides, you must consider both roots:
x = ±√400
x = ±20
So x = 20 or x = -20.
If you only write x = √400 = 20, you've lost half the solution. That's a points-losing mistake on a test — and in real modeling, it can mean missing a valid physical scenario Worth keeping that in mind..
Distance and Geometry
Here's a classic trap. Now, the distance between two points uses squares and square roots. Here's the thing — distance is always non-negative. So when you derive the distance formula, you're using the principal root Which is the point..
But if you're working with vectors or displacement, direction matters. Negative roots show up naturally there.
Quadratic Formula
The quadratic formula has a ± built right in:
x = [-b ± √(b² - 4ac)] / 2a
That ± is exactly the principal and negative square roots of the discriminant. You can't use the formula correctly without understanding both Most people skip this — try not to..
How to Find Square Roots (Without a Calculator)
Since we're here, let's talk about how you'd figure out √400 if you didn't just know it.
Perfect Squares You Should Know
Memorize these. It saves so much time:
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
20² = 400
21² = 441
22² = 484
23² = 529
24² = 576
25² = 625
See 400 in that list? 20². Done It's one of those things that adds up. Worth knowing..
Estimation for Non-Perfect Squares
What if it was √450?
You know 20² = 400 and 21² = 441 and 22² = 484.
So √450 is between 21 and 22. Closer to 21.
That kind of estimation is a superpower on multiple-choice tests.
Prime Factorization Method
For larger perfect squares, break it down:
400 = 4 × 100
= 2² × 10²
= (2 × 10)²
= 20²
So √400 = 20.
This works beautifully for numbers like 1764 (42²) or 3136 (56²) — numbers you'd never guess but can factor.
Common Mistakes (And How to Avoid Them)
I've graded a lot of math papers. These errors show up constantly But it adds up..
Mistake 1: Thinking √400 = ±20
No. √400 = 20.
Because of that, the equation x² = 400 has solutions x = ±20. The expression √400 has one value: 20.
This distinction is pedantic until it isn't. Day to day, in programming, Math. That's why sqrt(400) returns 20. In calculus, when you're dealing with derivatives of √x, it matters. Never -20 And it works..
Mist
Mistake 2: Forgetting Both Roots When Solving Equations
When you have x² = 400, taking the square root of both sides means:
√(x²) = √400
|x| = 20
x = ±20
Or more directly: if x² = 400, then x = ±√400 = ±20 Less friction, more output..
The absolute value approach shows why we need both signs—squaring eliminates sign information, so we must recover it.
Mistake 3: Applying Square Roots to Negative Numbers
You cannot take the square root of a negative number in the real number system. If you encounter √(-400), you've made an earlier error. The expression is undefined in real numbers.
In advanced mathematics, you'll learn about imaginary numbers where √(-1) = i, making √(-400) = 20i. But that's a topic for later.
Mistake 4: Confusing √(a²) with a
Remember: √(a²) = |a|, not just a And it works..
√(20²) = 20 ✓
√((-20)²) = √400 = 20 ≠ -20 ✗
The square root function always returns the non-negative value.
Why This Matters Beyond the Classroom
Understanding square roots deeply isn't just about passing algebra. It's about building mathematical thinking that scales.
In physics, when you solve for velocity from kinetic energy (v = √(2KE/m)), you need the positive root because velocity magnitude is positive. But in projectile motion, you might need both roots to find when an object hits the ground Practical, not theoretical..
In finance, when calculating compound interest rates, you might take square roots to solve for time periods. Missing one solution could mean missing an investment opportunity.
In computer graphics, distance calculations and vector operations rely on square roots constantly. Getting the sign wrong breaks rendering algorithms The details matter here..
Building Your Mathematical Foundation
The key insight is this: the principal square root is the "official" answer to "what is √400?" But mathematics is about relationships and solutions, not just single answers.
When you solve x² = 400, you're asking "what numbers, when squared, give 400?" Both 20 and -20 work. The ± symbol is your way of capturing all solutions Easy to understand, harder to ignore. Turns out it matters..
Think of it like this: the square root symbol √ is a function that gives you one output. The process of solving an equation like x² = 400 is about finding all inputs that produce a given output Simple, but easy to overlook..
Your Next Steps
Master these fundamentals, and you'll breeze through:
- Quadratic equations
- Trigonometric identities
- Calculus derivatives involving roots
- Physics equations
- Engineering formulas
Start by memorizing those perfect squares up to 25². Practice factoring larger numbers. Solve lots of x² = c problems.
Most importantly, internalize the difference between evaluating √x (one answer) and solving x² = c (two answers). This distinction will save you points and confusion for years to come It's one of those things that adds up..
Mathematics rewards precision in language and logic. Get these basics rock solid now, and you'll have the foundation to tackle whatever math comes next Easy to understand, harder to ignore..
