Learn Algebra
Master algebra fundamentals, solving techniques, and avoid common mistakes with our comprehensive guides.
📚 Algebra Fundamentals
Algebraic Properties
a + b = b + a
a × b = b × a
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
a(b + c) = ab + ac
a + 0 = a
a × 1 = a
Order of Operations (PEMDAS)
Evaluate expressions inside parentheses first
Calculate powers and roots
From left to right
From left to right
🔧 Solving Techniques
Factoring Guide
Greatest Common Factor (GCF)
Find the largest factor common to all terms.
Example: 6x² + 9x = 3x(2x + 3)
Difference of Squares
Pattern: a² - b² = (a + b)(a - b)
Example: x² - 16 = (x + 4)(x - 4)
Trinomial Factoring (x² + bx + c)
Find two numbers that multiply to c and add to b.
Example: x² + 5x + 6 = (x + 2)(x + 3)
Because 2 × 3 = 6 and 2 + 3 = 5
Perfect Square Trinomials
Pattern: a² + 2ab + b² = (a + b)²
Example: x² + 6x + 9 = (x + 3)²
Completing the Square
A technique for solving quadratic equations by creating a perfect square trinomial.
Steps:
- Move constant term to the right side
- Take half of the coefficient of x, then square it
- Add this value to both sides
- Factor the left side as a perfect square
- Take the square root of both sides
- Solve for x
Example: x² + 6x + 5 = 0
1. x² + 6x = -5
2. Half of 6 is 3, and 3² = 9
3. x² + 6x + 9 = -5 + 9
4. (x + 3)² = 4
5. x + 3 = ±2
6. x = -3 ± 2, so x = -1 or x = -5
⚠️ Common Mistakes to Avoid
❌ Distributing Incorrectly
Wrong: (x + 3)² = x² + 9
✓ Correct: (x + 3)² = x² + 6x + 9
Remember: (a + b)² = a² + 2ab + b²
❌ Canceling Terms Incorrectly
Wrong: (x + 3)/(x + 5) = 3/5
✓ Correct: You can't cancel terms that are being added
Only cancel factors, not terms
❌ Forgetting Negative Signs
Wrong: -(x - 3) = -x - 3
✓ Correct: -(x - 3) = -x + 3
Distribute the negative to all terms
❌ Division by Zero
Wrong: x/0 = undefined (but treating it like 0)
✓ Correct: Division by zero is undefined
Always check denominators aren't zero
❌ Ignoring Order of Operations
Wrong: 2 + 3 × 4 = 20
✓ Correct: 2 + 3 × 4 = 2 + 12 = 14
Multiplication before addition (PEMDAS)
✍️ Practice Problems
Practice makes perfect! Try these problems organized by difficulty level.
🟢 Easy
1. Solve for x: 2x + 5 = 13
2. Simplify: 3x + 2x - 5x
3. Factor: x² - 9
🟡 Medium
1. Solve for x: x² - 5x + 6 = 0
2. Simplify: (2x + 3)(x - 4)
3. Solve system: 2x + y = 10, x - y = 2
🔴 Hard
1. Solve: 2x² - 7x + 3 = 0
2. Simplify: (x² - 4)/(x + 2)
3. Complete the square: x² + 8x - 1 = 0
💡 Tip: Try solving these on paper first, then check your answers using our solvers!
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