Quadratic Equations - Standard Form and Setup
1️⃣ Understand the standard form of quadratic equations
2️⃣ Learn to identify coefficients A, B, and C
3️⃣ Master rearranging equations to standard form
4️⃣ Handle special cases with missing terms
5️⃣ Prepare equations for the quadratic formula
Before we can solve quadratic equations using the quadratic formula, we must first arrange them in the correct standard form. This is a crucial step that ensures we can properly identify the coefficients.
Standard Form
ax² + bx + c = 0
Where A ≠ 0, and the equation equals zero
Coefficient Identification
- A = coefficient of x² (must not equal zero)
- B = coefficient of x (can be zero)
- C = constant term (can be zero)
- The entire equation must equal zero
Very important: Sometimes the equation has a number on the right side. We must move it to the left and change its sign so the entire equation equals zero.
Rearranging Examples
Example 1: Moving the constant
Given: x² + 3x = 5
Move 5 to left: x² + 3x - 5 = 0
Now: A = 1, B = 3, C = -5
Example 2: More complex rearranging
Given: 2x² - x = 4
Move 4 to left: 2x² - x - 4 = 0
Now: A = 2, B = -1, C = -4
Sometimes there's no constant number in the equation. In this case, C = 0, and we can still use the quadratic formula.
Missing Constant Examples
Example 1
x² + x = 0
A = 1, B = 1, C = 0
Example 2
3x² - 2x = 0
A = 3, B = -2, C = 0
Sometimes there's no x term (linear term). In this case, B = 0, and we can still use the quadratic formula.
Missing Linear Term Examples
Example 1: Pure quadratic
Given: x² + 9 = 0
Coefficients: A = 1, B = 0, C = 9
Example 2: With different coefficient
Given: 2x² - 8 = 0
Coefficients: A = 2, B = 0, C = -8
Follow this systematic approach to prepare any quadratic equation for the quadratic formula.
Setup Process
- Move all terms to one side so the equation equals zero
- Arrange in descending order: x² term, x term, constant
- Identify A, B, C: Pay attention to signs
- Handle missing terms: Use 0 for missing coefficients
- Apply the quadratic formula with correct values
Complete Setup Examples
Example 1: Standard rearrangement
Given: x² + 2x = 3
Step 1: Move 3 to left → x² + 2x - 3 = 0
Step 2: Already in correct order
Step 3: A = 1, B = 2, C = -3
Ready for quadratic formula!
Example 2: Missing linear term
Given: 3x² = 12
Step 1: Move 12 to left → 3x² - 12 = 0
Step 2: No x term present
Step 3: A = 3, B = 0, C = -12
Ready for quadratic formula!
Example 3: Missing constant term
Given: 2x² + 5x = 0
Step 1: Already equals zero
Step 2: Already in correct order
Step 3: A = 2, B = 5, C = 0
Ready for quadratic formula!
🧠 Key Reminders
Critical Points to Remember
- Always set equation = 0 before identifying coefficients
- Pay attention to signs when moving terms
- Missing terms = 0 coefficient (not undefined)
- A ≠ 0 always (otherwise it's not quadratic)
- B and C can be zero in special cases
- Order matters: arrange as ax² + bx + c = 0
🎯 Common Mistakes to Avoid
- Forgetting to move terms: Don't leave numbers on the right side
- Sign errors: When moving terms, change their signs
- Missing coefficient confusion: No term means coefficient = 0
- Wrong order: Always arrange in descending powers of x
- A = 0 error: If A = 0, it's not a quadratic equation