The search for 'X' is the backbone of all mathematics. From calculating the tip at a restaurant to sending rockets to Mars, equation solving is the tool that turns unknowns into knowns. It is the art of undoing mathematical complexity.

1. The Golden Rule of Algebra

An equation is a balance scale. The equals sign (=) means the two sides weigh exactly the same.
Rule: Whatever you do to one side, you MUST do to the other.

If you subtract 5 from the left, the scale tips. You must subtract 5 from the right to re-balance it. This simple physical intuition is the proof behind every algebraic step.

2. Linear Equations (Isolation)

For simple equations like 3x + 5 = 20, we use the Onion Method (Peeling steps away).

• Goal: Isolate X.
• Reverse PEMDAS: When solving, we work backwards (SADMEP). Undo addition/subtraction first, then multiplication/division.
• Step 1: Undo the +5. Subtract 5 from both sides. (3x = 15).
• Step 2: Undo the *3. Divide by 3. (x = 5).

3. Quadratic Equations (The Monster)

What if x is squared (x²)? This changes everything because squaring destroys information ((-3)² and 3² are both 9). You generally have 3 ways to solve them:

Method A: Factoring
Find two numbers that multiply to C and add to B. This is fast but doesn't always work.

Method B: The Quadratic Formula
The nuclear option. It works on EVERY quadratic equation.

The part inside the root (b² - 4ac) is called the Discriminant.
• Positive = 2 Real Answers.
• Zero = 1 Real Answer.
• Negative = 2 Imaginary Answers.

4. Exponential Equations

How do you solve 2^x = 10? You can't just divide by 2. This is where Logarithms come in. Logarithms are the "undo" button for exponents.

Take log of both sides:
x * log(2) = log(10)
x = log(10) / log(2) ≈ 3.32.

5. Common Mistakes

• Dividing by Zero: Never divide by 'x' unless you know x is not 0. You might lose a solution! Factor instead.
• Square Root errors: Remember that √(9) is ±3, not just +3. If you forget the minus, you lose half the answer.

6. Real World Uses

• Finance: Compound interest formulas solve for 't' (time) to see how long until you become a millionaire.
• Medicine: Dosage calculations use algebra to determine how much medicine to give based on body weight.

Conclusion

Equation solving is not about memorizing tricks; it's about learning the logic of reversibility. Once you know how to reverse every operation (add/sub, mul/div, exp/log), no variable can hide from you.