Algebra · StudentHub Lesson
Quadratic Equations
Solve quadratic equations by factoring, using the difference of squares and the zero product rule.
What you will learn
- Recognize the standard form of a quadratic equation
- Factor simple quadratics and difference of squares
- Apply the zero product property
- Solve for two possible roots
- Verify solutions by substitution
Watch the lesson
How To Solve Quadratic Equations By Factoring - Quick & Simple! · The Organic Chemistry Tutor
Watch on YouTubeTopic notes
Main Idea
A quadratic equation has the form ax^2 + bx + c = 0. Many can be solved by factoring into two binomials and using the zero product rule.
Key Concepts
- Difference of squares: a^2 - b^2 = (a+b)(a-b)
- Factoring trinomials: find two numbers that multiply to c and add to b
- Zero product rule: if two factors multiply to zero, at least one must be zero
Definitions
- Quadratic equation: an equation with a squared variable term
- Root/Solution: value(s) of x that satisfy the equation
- Factoring: rewriting an expression as a product of simpler expressions
Formulas
- Standard form: ax^2 + bx + c = 0
- Difference of squares: x^2 - k^2 = (x-k)(x+k)
Examples
- x^2 - 49 = 0 → (x-7)(x+7) = 0 → x = 7 or x = -7
- x^2 + 5x + 6 = 0 → (x+2)(x+3) = 0 → x = -2 or x = -3
Common Mistakes
- Forgetting a quadratic can have two solutions
- Sign errors when factoring
- Applying the zero product rule before setting the equation equal to zero
Key concepts
Important terms
- Quadratic
- A polynomial equation with highest power 2.
- Factoring
- Writing an expression as a product of factors.
- Root
- A solution to the equation.
Worked examples
Problem
Solve x^2 - 49 = 0
- 1. Recognize difference of squares
- 2. Factor: (x-7)(x+7) = 0
- 3. Set each factor to zero
- 4. x = 7 or x = -7
Answer: x = 7 or x = -7
Problem
Solve x^2 + 5x + 6 = 0
- 1. Find two numbers multiplying to 6 and adding to 5: 2 and 3
- 2. Factor: (x+2)(x+3) = 0
- 3. Set each factor to zero
- 4. x = -2 or x = -3
Answer: x = -2 or x = -3
Quick revision
- Set the equation equal to zero first
- Look for the difference of squares pattern
- Find factor pairs for trinomials
- Zero product rule gives two solutions
- Always check both roots
- Not all quadratics factor nicely
Check your understanding
Question 1 · Multiple choice
Factor x² - 9
Question 2 · Multiple choice
Solve x² - 4 = 0
Question 3 · Short answer
Solve x² + 7x + 12 = 0
Question 4 · Short answer
Solve x² - x - 6 = 0
Question 5 · True or false
A quadratic equation can have two different solutions.
Done with Quadratic Equations?
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