Solve equations involving squares and square roots

Key Notes :

🌟 Solve Equations Involving Squares and Square Roots

🔹 What is a Square ?

The square of a number is the result of multiplying the number by itself.

👉 Example:

• 4²=4×4=16

🔹 What is a Square Root ?

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

👉 Example:

• √25=5 because 5×5=25

🧮 Solving Equations Involving Squares

✴️ Example 1:

Solve x²=49

Step 1: Take the square root of both sides

x=±√49​

Step 2: Simplify

x=±7

✅ Final Answer: x=7 or x=−7

🧩 Solving Equations Involving Square Roots

✴️ Example 2:

Solve √x=8

Step 1: Square both sides to remove the square root

(√x)²=8²

Step 2: Simplify

x=64

✅ Final Answer: x=64

🧠 Remember These Important Rules

✅ When squaring both sides, check for extraneous solutions (false answers).

✅ A square equation can have two solutions — one positive and one negative.

✅ A square root equation has only non-negative roots (since square root of a number is defined as the principal or positive root).

💡 Common Forms and How to Solve

Equation Type	Example	Solution Method
x2=a	x2=36	x=±√36=±6
√x=a	√x=9	x=a2=81
(x−k)2=a	(x−3)2=16	x−3=±4 ⇒ x=7 or −1
√x+k=a	√x+2=5	x+2=25 ⇒ x=23

⚠️ Check Your Answer

Always substitute your solution back into the original equation to verify it satisfies the equation.

🎯 Key Takeaways

🌈 Squaring removes square roots.

🌈 Taking square roots removes squares.

🌈 Every square equation may have two solutions.

🌈 Always check for valid answers after squaring.

Learn with an example

Let’s practice!