What is an Equation?
An equation is a mathematical statement that asserts the equality of two expressions. The goal of solving an equation is to find the value(s) of the variable(s) that make the statement true. These values are called solutions or roots of the equation.
Algebra provides systematic methods for solving equations, replacing guesswork with logical procedures that work for any equation of a given type. Understanding these methods builds a foundation for more advanced mathematics and real-world problem-solving.
The Golden Rule of Equations
The fundamental principle of solving equations is: whatever you do to one side, you must do to the other. This preserves the equality while transforming the equation into a simpler form.
The Balance Analogy
Think of an equation as a balanced scale. If you add weight to one side, you must add the same weight to the other side to keep it balanced. The same applies to subtracting, multiplying, or dividing.
Solving Linear Equations
A linear equation has the variable raised to the first power only (no x², x³, etc.). The standard form is ax + b = c, where a, b, and c are constants.
Step-by-Step Method
- Simplify both sides: Distribute and combine like terms
- Move variables to one side: Add or subtract terms to get all x terms on one side
- Move constants to the other side: Add or subtract to isolate the variable term
- Divide to solve: Divide both sides by the coefficient of x
- Check your answer: Substitute back into the original equation
Example 1: Simple Linear Equation
Solve: 3x + 7 = 22
Step 1: Subtract 7 from both sides: 3x = 15
Step 2: Divide both sides by 3: x = 5
Check: 3(5) + 7 = 15 + 7 = 22 ✓
Example 2: Variables on Both Sides
Solve: 5x - 3 = 2x + 12
Step 1: Subtract 2x from both sides: 3x - 3 = 12
Step 2: Add 3 to both sides: 3x = 15
Step 3: Divide by 3: x = 5
Check: 5(5) - 3 = 22 and 2(5) + 12 = 22 ✓
Equations with Fractions
When equations contain fractions, multiply both sides by the least common denominator (LCD) to eliminate fractions first.
Example 3: Equation with Fractions
Solve: x/3 + x/4 = 7
LCD of 3 and 4 is 12
Multiply all terms by 12: 4x + 3x = 84
Combine: 7x = 84
Divide: x = 12
Solving Quadratic Equations
A quadratic equation has the variable raised to the second power. The standard form is ax² + bx + c = 0. Quadratic equations can have two solutions, one solution, or no real solutions.
Method 1: Factoring
If a quadratic expression can be factored, factoring is often the quickest solution method.
Example 4: Solving by Factoring
Solve: x² + 5x + 6 = 0
Factor: (x + 2)(x + 3) = 0
Set each factor equal to zero:
x + 2 = 0 → x = -2
x + 3 = 0 → x = -3
Solutions: x = -2 or x = -3
Factoring Tip
For x² + bx + c = 0, look for two numbers that multiply to c and add to b. Those numbers become the constants in your factors.
Method 2: The Quadratic Formula
The quadratic formula works for ALL quadratic equations, including those that cannot be factored.
Example 5: Using the Quadratic Formula
Solve: 2x² - 7x + 3 = 0
Here: a = 2, b = -7, c = 3
x = (7 ± √(49 - 24)) / 4
x = (7 ± √25) / 4 = (7 ± 5) / 4
x = 12/4 = 3 or x = 2/4 = 0.5
The Discriminant
The expression under the square root, b² - 4ac, is called the discriminant. It tells us about the nature of the solutions:
- If b² - 4ac > 0: Two distinct real solutions
- If b² - 4ac = 0: One repeated real solution
- If b² - 4ac < 0: No real solutions (complex solutions)
Method 3: Completing the Square
This method rewrites the quadratic as a perfect square trinomial. While more involved, it's essential for deriving the quadratic formula and understanding conic sections.
Example 6: Completing the Square
Solve: x² + 6x + 5 = 0
Move constant: x² + 6x = -5
Add (6/2)² = 9 to both sides: x² + 6x + 9 = 4
Factor: (x + 3)² = 4
Take square root: x + 3 = ±2
Solve: x = -1 or x = -5
Common Mistakes to Avoid
- Sign errors: Be careful when moving terms across the equals sign—signs change!
- Not distributing completely: When multiplying, distribute to ALL terms
- Forgetting to check: Always verify solutions in the original equation
- Dividing by the variable: This can eliminate valid solutions (like x = 0)
- Missing the ± in square roots: Remember both positive and negative roots
Critical Warning
Never divide both sides by an expression containing the variable. This can eliminate valid solutions. Instead, move all terms to one side and factor.
Practice Strategy
To master equation solving:
- Start with simple equations and gradually increase complexity
- Write every step—skipping steps leads to errors
- Always check your answers by substitution
- Identify equation types before solving to choose the best method
- Practice recognizing factorable quadratics quickly