Solved Problems: Factoring Quadratic Equations Step-by-Step

📚 What is Factoring Quadratic Equations?

Factoring a quadratic equation involves expressing it as a product of two binomials. A quadratic equation is generally in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. Factoring simplifies solving for the roots (or solutions) of the equation.

📜 History and Background

The concept of solving quadratic equations dates back to ancient civilizations, including the Babylonians and Egyptians. They used geometric and algebraic methods to find solutions. The systematic approach we use today evolved over centuries, with contributions from mathematicians worldwide.

🔑 Key Principles of Factoring

• 🔍 Identify the Quadratic Form: Recognize the equation in the form $ax^2 + bx + c = 0$.
• 💡 Find Factors of 'ac': Determine two numbers that multiply to $ac$ and add up to $b$.
• 📝 Rewrite the Middle Term: Replace $bx$ with the two terms found in the previous step.
• ➗ Factor by Grouping: Group the terms and factor out the greatest common factor (GCF).
• ✅ Verify the Factors: Ensure the product of the binomials equals the original quadratic equation.

➗ Example 1: Simple Factoring

Solve $x^2 + 5x + 6 = 0$

1. Find two numbers that multiply to 6 and add to 5: These numbers are 2 and 3.
2. Rewrite the equation: $x^2 + 2x + 3x + 6 = 0$
3. Factor by grouping: $x(x + 2) + 3(x + 2) = 0$
4. Factor out the common binomial: $(x + 2)(x + 3) = 0$
5. Solutions: $x = -2$ or $x = -3$

➕ Example 2: Factoring with a Leading Coefficient

Solve $2x^2 + 7x + 3 = 0$

1. Find two numbers that multiply to $2*3 = 6$ and add to 7: These numbers are 1 and 6.
2. Rewrite the equation: $2x^2 + x + 6x + 3 = 0$
3. Factor by grouping: $x(2x + 1) + 3(2x + 1) = 0$
4. Factor out the common binomial: $(2x + 1)(x + 3) = 0$
5. Solutions: $x = -\frac{1}{2}$ or $x = -3$

➖ Example 3: Factoring Difference of Squares

Solve $x^2 - 9 = 0$

1. Recognize this as a difference of squares: $x^2 - 3^2 = 0$
2. Apply the formula $a^2 - b^2 = (a + b)(a - b)$: $(x + 3)(x - 3) = 0$
3. Solutions: $x = 3$ or $x = -3$

➗ Example 4: Perfect Square Trinomial

Solve $x^2 + 6x + 9 = 0$

1. Recognize this as a perfect square trinomial: $(x + 3)^2 = 0$
2. Factor: $(x + 3)(x + 3) = 0$
3. Solution: $x = -3$

➕ Example 5: Factoring with GCF

Solve $3x^2 + 12x + 9 = 0$

1. Factor out the GCF (3): $3(x^2 + 4x + 3) = 0$
2. Factor the quadratic: $3(x + 1)(x + 3) = 0$
3. Solutions: $x = -1$ or $x = -3$

➖ Example 6: Factoring with Negative Coefficients

Solve $-x^2 + 5x - 6 = 0$

1. Multiply by -1: $x^2 - 5x + 6 = 0$
2. Factor: $(x - 2)(x - 3) = 0$
3. Solutions: $x = 2$ or $x = 3$

➗ Example 7: Advanced Factoring

Solve $4x^2 - 12x + 9 = 0$

1. Recognize as a perfect square trinomial: $(2x - 3)^2 = 0$
2. Factor: $(2x - 3)(2x - 3) = 0$
3. Solution: $x = \frac{3}{2}$

💡 Conclusion

Factoring quadratic equations is a fundamental skill in algebra. By understanding the key principles and practicing various examples, you can master this technique. Remember to always look for the greatest common factor first and recognize special patterns like the difference of squares or perfect square trinomials. Happy factoring!