The Quadratic Formula - Algebra-1

1. Fundamental Concepts

Definition: The quadratic formula is a method used to solve quadratic equations of the form $$ax^2 + bx + c = 0$$ , where $$a \neq 0$$ .
Formula: The solutions are given by $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$ .
Discriminant: The expression under the square root, $$b^2 - 4ac$$ , determines the nature of the roots (real, complex, distinct, or repeated).

2. Key Concepts

Basic Rule: $$ax^2 + bx + c = 0 \Rightarrow x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$

Degree Preservation: The highest degree in the result matches input

Application: Used to find the roots of quadratic equations in various fields including physics and engineering

3. Examples

Example 1 (Basic)

Problem: Solve $$x^2 - 5x + 6 = 0$$ using the quadratic formula.

Step-by-Step Solution:

Identify coefficients: $$a = 1$$ , $$b = -5$$ , $$c = 6$$ .
Substitute into the formula: $$x = \frac{{-(-5) \pm \sqrt{{(-5)^2 - 4(1)(6)}}}}{{2(1)}}$$ .
Simplify: $$x = \frac{{5 \pm \sqrt{{25 - 24}}}}{{2}} = \frac{{5 \pm \sqrt{{1}}}}{{2}}$$ .
Final solutions: $$x = \frac{{5 + 1}}{{2}} = 3$$ and $$x = \frac{{5 - 1}}{{2}} = 2$$ .

Validation: Substitute $$x = 3$$ and $$x = 2$$ into the original equation → Original: $$3^2 - 5(3) + 6 = 0$$ ; Simplified: $$9 - 15 + 6 = 0$$ ✓

Example 2 (Intermediate)

Problem: Solve $$2x^2 + 3x - 2 = 0$$ using the quadratic formula.

Step-by-Step Solution:

Identify coefficients: $$a = 2$$ , $$b = 3$$ , $$c = -2$$ .
Substitute into the formula: $$x = \frac{{-3 \pm \sqrt{{3^2 - 4(2)(-2)}}}}{{2(2)}}$$ .
Simplify: $$x = \frac{{-3 \pm \sqrt{{9 + 16}}}}{{4}} = \frac{{-3 \pm \sqrt{{25}}}}{{4}}$$ .
Final solutions: $$x = \frac{{-3 + 5}}{{4}} = \frac{{1}}{{2}}$$ and $$x = \frac{{-3 - 5}}{{4}} = -2$$ .

Validation: Substitute $$x = \frac{{1}}{{2}}$$ and $$x = -2$$ into the original equation → Original: $$2\left(\frac{{1}}{{2}}\right)^2 + 3\left(\frac{{1}}{{2}}\right) - 2 = 0$$ ; Simplified: $$\frac{{1}}{{2}} + \frac{{3}}{{2}} - 2 = 0$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use graphs to visualize the quadratic function and its roots.
Error-Proofing: Double-check the values of $$a$$ , $$b$$ , and $$c$$ before substituting into the formula.
Concept Reinforcement: Practice with different types of quadratic equations to reinforce understanding of the formula.