How to Solve Linear and Quadratic Equations — Step-by-Step Methods

Understanding Equation Types and Solution Goals

An equation is a mathematical statement that two expressions are equal. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true—these are called solutions or roots.

1. Linear Equations (First Degree)

A linear equation has the form:
ax + b = 0
where a and b are constants, and a ≠ 0. It has exactly one solution.

Solution Method: Isolate the variable using inverse operations (addition/subtraction, multiplication/division).

Example:
3x – 7 = 8
→ Add 7: 3x = 15
→ Divide by 3: x = 5

2. Quadratic Equations (Second Degree)

A quadratic equation has the form:
ax² + bx + c = 0
where a ≠ 0. It can have 0, 1, or 2 real solutions, depending on the discriminant.

Method 1: Factoring

Best when the quadratic factors neatly into two binomials.

Example:
x² – 5x + 6 = 0
→ Factor: (x – 2)(x – 3) = 0
→ Set each factor to zero: x = 2 or x = 3

Method 2: Quadratic Formula

The universal method that works for all quadratics: x = [-b ± √(b² – 4ac)] / (2a)

The discriminant D = b² – 4ac determines the nature of the roots:

• D > 0: Two distinct real roots
• D = 0: One real root (repeated)
• D below 0: Two complex conjugate roots

Example:
2x² + 4x – 6 = 0
→ a=2, b=4, c=–6
→ D = 16 + 48 = 64
→ x = [–4 ± 8]/4 → x = 1 or x = –3

Method 3: Completing the Square

Useful for deriving the quadratic formula and solving equations not easily factored.

Steps:

1. Move constant to right: x² + 6x = –5
2. Add (b/2)² to both sides: x² + 6x + 9 = 4
3. Write as square: (x + 3)² = 4
4. Take square root: x + 3 = ±2
5. Solve: x = –1 or x = –5

3. Systems of Linear Equations

A system involves two or more equations with multiple variables. The solution is the point where their graphs intersect.

Substitution Method

1. Solve one equation for one variable.
2. Substitute into the other equation.
3. Solve and back-substitute.

Example:
y = 2x + 1
3x + y = 11
→ Substitute: 3x + (2x + 1) = 11 → 5x = 10 → x = 2
→ y = 5

Elimination Method

1. Multiply equations to align coefficients.
2. Add/subtract to eliminate one variable.
3. Solve and back-substitute.

Example:
2x + 3y = 13
–x + 4y = 5
→ Multiply second by 2: –2x + 8y = 10
→ Add: 11y = 23 → y = 23/11
→ x = 37/11

Pro Tips & Common Mistakes

• Always simplify first: Combine like terms and move all terms to one side before solving.
• Check your solutions: Plug answers back into the original equation to verify.
• Use fractions for precision: Avoid decimals until the final step to prevent rounding errors.
• Watch signs: A common error is mishandling negative signs during distribution or transposition.
• Know when to use each method:
  • Factoring: Fast for simple quadratics
  • Quadratic formula: Reliable for all cases
  • Completing the square: Needed for vertex form or conic sections
• For systems: Use elimination when coefficients are easily aligned; substitution when one equation is already solved for a variable.

Practical Applications

• Finance: Break-even analysis (Revenue = Cost)
• Physics: Kinematic equations (d = vt + ½at²)
• Engineering: Circuit analysis (Ohm’s Law systems)
• Economics: Supply and demand equilibrium
• Computer Science: Algorithm complexity and optimization
• Everyday life: Budgeting, recipe scaling, travel planning