How to Solve Quadratic Equations Step by Step

Quadratic equations are fundamental to algebra and appear everywhere in mathematics, physics, and engineering. Whether you're a student struggling with homework or preparing for standardized tests, mastering these equations is essential. In this comprehensive guide, we'll break down three proven methods to solve any quadratic equation.

What is a Quadratic Equation?

A quadratic equation is any equation that can be written in the standard form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The variable x represents an unknown value we're trying to find.

Examples of Quadratic Equations:

• x² + 5x + 6 = 0
• 2x² - 7x + 3 = 0
• x² - 16 = 0

Method 1: Factoring

Factoring is often the quickest method when it works. The goal is to rewrite the equation as a product of two binomials.

Step-by-Step Process:

1. Ensure the equation is in standard form (ax² + bx + c = 0)
2. Find two numbers that multiply to give ac and add to give b
3. Rewrite the middle term using these two numbers
4. Factor by grouping
5. Set each factor equal to zero and solve

Example: Solve x² + 5x + 6 = 0

1. We need two numbers that multiply to 6 and add to 5: 2 and 3
2. Factor: (x + 2)(x + 3) = 0
3. Set each factor to zero:
   • x + 2 = 0 → x = -2
   • x + 3 = 0 → x = -3
4. Solution: x = -2 or x = -3

Method 2: Completing the Square

This method works for all quadratic equations and helps understand the derivation of the quadratic formula.

Step-by-Step Process:

1. Move the constant term to the right side
2. Divide all terms by a (if a ≠ 1)
3. Take half of the b coefficient, square it, and add to both sides
4. Factor the left side as a perfect square
5. Take the square root of both sides
6. Solve for x

Example: Solve x² + 6x + 5 = 0

1. x² + 6x = -5
2. Take half of 6 (which is 3), square it (9), add to both sides: x² + 6x + 9 = -5 + 9
3. Factor: (x + 3)² = 4
4. Take square root: x + 3 = ±2
5. Solve: x = -3 + 2 = -1 or x = -3 - 2 = -5
6. Solution: x = -1 or x = -5

Method 3: The Quadratic Formula

The quadratic formula is the most powerful method because it works for every quadratic equation, even when factoring is impossible.

The Formula:

x = (-b ± √(b² - 4ac)) / 2a

Step-by-Step Process:

1. Identify values of a, b, and c from the standard form
2. Calculate the discriminant: b² - 4ac
3. Substitute values into the formula
4. Simplify to find the solutions

Example: Solve 2x² + 7x + 3 = 0

1. Identify: a = 2, b = 7, c = 3
2. Calculate discriminant: 7² - 4(2)(3) = 49 - 24 = 25
3. Apply formula: x = (-7 ± √25) / (2 × 2)
4. Simplify: x = (-7 ± 5) / 4
5. Two solutions:
   • x = (-7 + 5) / 4 = -2/4 = -0.5
   • x = (-7 - 5) / 4 = -12/4 = -3
6. Solution: x = -0.5 or x = -3

Understanding the Discriminant

The discriminant (b² - 4ac) tells us important information about the solutions:

• Positive discriminant: Two distinct real solutions
• Zero discriminant: One repeated real solution
• Negative discriminant: Two complex solutions (no real solutions)

Which Method Should You Use?

Choose your method based on the equation:

• Factoring: Best for simple equations with integer solutions
• Completing the Square: Good for understanding and when a = 1
• Quadratic Formula: Universal solution that works for all equations

Common Mistakes to Avoid

1. Forgetting the ± sign in the quadratic formula
2. Arithmetic errors when calculating the discriminant
3. Not simplifying final answers
4. Missing solutions by only considering positive square roots

Practice Problems

Try solving these on your own:

1. x² - 7x + 12 = 0
2. 3x² + 11x - 4 = 0
3. x² + 4x + 4 = 0

Conclusion

Mastering quadratic equations takes practice, but with these three methods in your toolkit, you're well-equipped to tackle any problem. Remember: factoring is fast when it works, completing the square helps build intuition, and the quadratic formula always works.

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