Positive and negative square roots

Key Notes :

🧮 Positive and Negative Square Roots

🌟 What is a Square Root?

The square root of a number is a value that, when multiplied by itself, gives the original number.

• 👉 Example: √9 = 3 because 3 × 3 = 9

➕ Positive Square Root

The positive square root is the non-negative value of a number’s square root.

• ✅ Example: √25 = +5

📘 We call it the principal square root.

• The symbol √ always represents the positive root unless stated otherwise.

➖ Negative Square Root

Every positive number also has a negative square root, because a negative times a negative equals a positive.

• 👉 Example: −5 × −5 = 25 → So, √25 = ±5

💡 Therefore, a positive number has two square roots:

√a=+number and −number

Example:

• √36 = ±6 → (that means +6 and −6)

🚫 Square Roots of Negative Numbers

The square root of a negative number is not a real number.

• ✖ Example: √(−9) = no real value, because no real number squared gives a negative result.
• (It belongs to imaginary numbers, represented using i in higher grades.)

📐 Perfect Squares and Their Roots

Number	Positive √	Negative √
4	+2	−2
9	+3	−3
16	+4	−4
25	+5	−5
36	+6	−6
49	+7	−7

🧠 Key Points to Remember

✅ Every positive number has two square roots (one positive, one negative).

✅ 0 has only one square root, which is 0.

✅ Negative numbers do not have real square roots.

✅ The principal (positive) square root is written with the √ symbol.

🌈 Examples

1️⃣ √49 = ±7

2️⃣ √64 = ±8

3️⃣ √0 = 0

4️⃣ √(−16) → Not a real number

💬 Shortcut Tip

• If x²=25, then x=±5
• 👉 Always remember to write both roots when solving equations like this!

Learn with an example

Let’s try some examples! ✍️