Complete Guide to the Quadratic Formula

What is the Quadratic Formula?

The quadratic formula is a mathematical formula used to solve quadratic equations of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

The Quadratic Formula:

x = (-b ± √(b² - 4ac)) / (2a)

When to Use the Quadratic Formula

Use the quadratic formula when:

The quadratic equation cannot be easily factored
You need exact solutions (not approximations)
The equation is in standard form ax² + bx + c = 0
Other methods like completing the square are too complex

Step-by-Step Solution Process

Step 1: Identify coefficients

From ax² + bx + c = 0, identify values of a, b, and c

Step 2: Calculate the discriminant

Find b² - 4ac to determine the nature of solutions

Step 3: Apply the formula

Substitute values into x = (-b ± √(b² - 4ac)) / (2a)

Step 4: Simplify

Calculate both solutions (+ and - versions)

Example Problem

Solve: 2x² + 5x - 3 = 0

Step 1: a = 2, b = 5, c = -3

Step 2: Discriminant = 5² - 4(2)(-3) = 25 + 24 = 49

Step 3: x = (-5 ± √49) / (2×2) = (-5 ± 7) / 4

Step 4: x₁ = (-5 + 7) / 4 = 1/2, x₂ = (-5 - 7) / 4 = -3

Solutions: x = 1/2 and x = -3

Understanding the Discriminant

b² - 4ac > 0

Two distinct real solutions

b² - 4ac = 0

One repeated real solution

b² - 4ac < 0

Two complex solutions

Common Mistakes to Avoid

Forgetting to write the equation in standard form first
Sign errors when identifying coefficients (especially with c)
Calculation errors in the discriminant
Forgetting the ± symbol (missing one solution)
Arithmetic mistakes in the final simplification

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