In this chapter, you will learn
- —Understand the standard form of quadratic equations
- —Solve quadratic equations by factorization
- —Solve quadratic equations by completing the square
- —Apply the quadratic formula for any quadratic equation
- —Determine the nature of roots using discriminant
- —Find the relationship between roots and coefficients
- —Form quadratic equations from given roots
- —Apply quadratic equations to real-world word problems
Standard Form and Introduction
Quadratic Equation: An equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
Standard Form: ax² + bx + c = 0
where:
• a ≠ 0 (if a = 0, it becomes linear, not quadratic)
• a = coefficient of x² (leading coefficient)
• b = coefficient of x
• c = constant term
Examples of Quadratic Equations:
- x² + 5x + 6 = 0 (a=1, b=5, c=6)
- 2x² - 3x - 2 = 0 (a=2, b=-3, c=-2)
- x² - 4 = 0 (a=1, b=0, c=-4)
- 3x² + 7x = 0 (a=3, b=7, c=0)
Not quadratic equations: x + 5 = 0 (linear), x³ + 2x = 0 (cubic)
Roots of Quadratic Equation: Values of x that satisfy the equation.
- A quadratic equation has at most 2 roots (real or complex)
- The roots can be equal, unequal, real, or complex
- If α and β are roots, then x = α and x = β satisfy the equation
Key Points:
- The degree of a quadratic equation is 2
- Leading coefficient a ≠ 0 is essential
- The equation represents a parabola when graphed
Exam Tip
Always ensure the equation is in standard form ax² + bx + c = 0 before identifying a, b, and c. Verify a ≠ 0.
Common Mistake
Students confuse quadratic with linear equations. Remember: quadratic has x², linear has only x. If a = 0, it's not quadratic.
Factorization Method
If a quadratic equation can be expressed as a product of two linear factors, we can find roots easily.
Method:
- Factorize ax² + bx + c as a(x - p)(x - q) or (x - p)(x - q) etc.
- Set each factor to zero: (x - p) = 0 and (x - q) = 0
- Solve to get x = p and x = q
Example 1: x² + 5x + 6 = 0
- Factorize: (x + 2)(x + 3) = 0
- So: x + 2 = 0 or x + 3 = 0
- Roots: x = -2 or x = -3
Example 2: 2x² - 5x + 3 = 0
- Factorize: (2x - 3)(x - 1) = 0
- So: 2x - 3 = 0 or x - 1 = 0
- Roots: x = 3/2 or x = 1
Tips for Factorization:
- For x² + bx + c, find two numbers that multiply to give c and add to give b
- For ax² + bx + c, use the ac-method or direct factorization
- Look for common factors first
When to Use:
- When the quadratic expression can be factorized easily
- Most efficient when a = 1 or when factors are obvious
- Not always possible for all quadratic equations
Key Points:
- Factorization works only when roots are rational
- Always verify roots by substitution
- This method is fastest when applicable
Exam Tip
Factorization is quickest. Look for patterns: x² - 4 = (x-2)(x+2), x² - 5x + 6 = (x-2)(x-3). Practice identifying factor pairs.
Common Mistake
Students make factorization errors. To find factors of ax² + bx + c: find numbers that multiply to 'ac' and add to 'b', then split the middle term.
Completing the Square Method
This method transforms the quadratic into a perfect square form, making it solvable.
Steps:
- Make coefficient of x² equal to 1 (divide entire equation by a if needed)
- Move constant term to right side
- Add (b/2)² to both sides to complete the square
- Express left side as perfect square and simplify right side
- Take square root of both sides
- Solve for x
Example: x² + 6x + 5 = 0
- Move constant: x² + 6x = -5
- Complete square: x² + 6x + (6/2)² = -5 + 9
- x² + 6x + 9 = 4
- Perfect square: (x + 3)² = 4
- Take square root: x + 3 = ±2
- Roots: x = -3 + 2 = -1 or x = -3 - 2 = -5
Example: 2x² + 8x - 10 = 0
- Divide by 2: x² + 4x - 5 = 0
- Move constant: x² + 4x = 5
- Complete square: x² + 4x + 4 = 5 + 4 = 9
- (x + 2)² = 9
- x + 2 = ±3
- Roots: x = 1 or x = -5
Advantage of This Method:
- Works for all quadratic equations
- Derives the quadratic formula
- Useful when factorization is difficult
Key Points:
- The term to add is always (b/2)²
- Add it to both sides to maintain equality
- Perfect square form: (x + p)² = q
Exam Tip
Remember: add (b/2)² to both sides. This is systematic and works always. Useful for finding exact roots even when they're irrational.
Common Mistake
Students forget to add (b/2)² to both sides or miscalculate (b/2)². Also, remember ±√ when taking square root. x = -b/2 ± √(answer).
Quadratic Formula
The quadratic formula directly gives roots for any quadratic equation without intermediate steps.
Quadratic Formula:
x = [-b ± √(b² - 4ac)] / (2a)
where the equation is ax² + bx + c = 0
Derivation (from completing the square):
- Starting: ax² + bx + c = 0
- Divide by a: x² + (b/a)x + c/a = 0
- Complete square: (x + b/(2a))² = b²/(4a²) - c/a
- Simplify: (x + b/(2a))² = (b² - 4ac)/(4a²)
- Take square root: x + b/(2a) = ±√(b² - 4ac)/(2a)
- Result: x = [-b ± √(b² - 4ac)] / (2a)
Example: 3x² - 5x + 2 = 0
- Here: a = 3, b = -5, c = 2
- Discriminant: b² - 4ac = 25 - 24 = 1
- x = [5 ± 1] / 6
- Roots: x = 6/6 = 1 or x = 4/6 = 2/3
Example: x² + 4x + 4 = 0
- Here: a = 1, b = 4, c = 4
- Discriminant: 16 - 16 = 0
- x = -4 / 2 = -2
- Equal roots: x = -2 (repeated root)
When to Use:
- Works for ANY quadratic equation
- Most reliable method for difficult quadratics
- Essential when factorization is impossible
Key Points:
- Always verify signs of a, b, c carefully
- Calculate discriminant (b² - 4ac) first to determine nature of roots
- Remember ± before the square root
- This is the universal method
Exam Tip
Master this formula completely. It works for all equations. Practice calculating discriminant first as it tells you about roots immediately.
Common Mistake
Sign errors are very common. Check: is it -b or +b? Is b negative or positive? Calculate b² - 4ac carefully, not b - 4ac. Remember ± gives two roots.
Nature of Roots and Discriminant
The discriminant (Δ = b² - 4ac) determines the nature of roots without solving the equation.
Discriminant: Δ = b² - 4ac
• If Δ > 0: Two distinct real roots (unequal)
• If Δ = 0: Two equal real roots (repeated root)
• If Δ < 0: No real roots (complex conjugate roots)
Analysis of Each Case:
- Δ > 0 (Two distinct real roots):
- Example: x² - 5x + 6 = 0 → Δ = 25 - 24 = 1 > 0
- Roots: x = 2 and x = 3 (different values)
- Parabola intersects x-axis at two points
- Δ = 0 (Equal real roots):
- Example: x² - 4x + 4 = 0 → Δ = 16 - 16 = 0
- Root: x = 2 (repeated, or x = 2, 2)
- Parabola touches x-axis at one point (vertex touches)
- Δ < 0 (No real roots):
- Example: x² + x + 1 = 0 → Δ = 1 - 4 = -3 < 0
- Roots are complex: x = (-1 ± i√3) / 2
- Parabola doesn't touch x-axis at all
When Δ is a Perfect Square:
- Roots are rational (can be expressed as fractions)
- Example: Δ = 9 → √Δ = 3 (rational roots possible)
- Example: Δ = 5 → √Δ = √5 (irrational roots)
Key Points:
- Calculate Δ first to understand the root nature
- If Δ < 0 and problem asks for real roots, answer is "No real roots"
- If Δ = 0, don't calculate ± separately; there's only one root value
- Discriminant saves time; no need to solve if Δ < 0
Exam Tip
Always calculate discriminant first. If Δ < 0 and question asks for real roots, say 'no real solutions'. If Δ = 0, explicitly state 'equal roots' or 'repeated root'.
Common Mistake
Students don't recognize that Δ = 0 means two equal roots (not one root). Write as x = α, α or say 'equal roots'. Also, Δ < 0 means NO REAL roots (not 'no roots').
Relationship Between Roots and Coefficients
If α and β are roots of ax² + bx + c = 0, there are important relationships with coefficients.
Vieta's Formulas:
• Sum of roots: α + β = -b/a
• Product of roots: αβ = c/a
Derivation:
- If α and β are roots: (x - α)(x - β) = 0
- Expand: x² - (α + β)x + αβ = 0
- Compare with x² + (b/a)x + (c/a) = 0
- α + β = -b/a and αβ = c/a
Example: 2x² - 7x + 3 = 0
- Here: a = 2, b = -7, c = 3
- Sum of roots: α + β = 7/2
- Product of roots: αβ = 3/2
- Solving: roots are 1/2 and 3
- Verification: 1/2 + 3 = 7/2 ✓ and (1/2)(3) = 3/2 ✓
Other Useful Relationships:
- (α - β)² = (α + β)² - 4αβ
- α² + β² = (α + β)² - 2αβ
- α/β + β/α = (α² + β²)/(αβ) = (α + β)² - 2αβ) / (αβ)
- 1/α + 1/β = (α + β) / (αβ)
Uses:
- Verify roots after solving
- Find relationships between roots without solving the equation
- Form a quadratic equation from given roots
Key Points:
- Sum = -b/a (note the negative sign)
- Product = c/a (direct ratio)
- These formulas work for any quadratic equation
Exam Tip
Remember the negative sign in sum formula: α + β = -b/a, not b/a. Use these formulas to verify your roots after solving.
Common Mistake
Most common: writing α + β = b/a instead of -b/a. Always remember the negative sign. Also, product = c/a, not a/c.
Forming Quadratic Equations from Roots
If roots α and β are given, we can construct the quadratic equation with those roots.
Methods to Form Quadratic Equation:
Method 1: If α and β are roots, then:
(x - α)(x - β) = 0
x² - (α + β)x + αβ = 0
Method 2: Using Vieta's formulas:
x² - (sum of roots)x + (product of roots) = 0
Example 1: Roots are 2 and 3
- Sum = 2 + 3 = 5
- Product = 2 × 3 = 6
- Equation: x² - 5x + 6 = 0
- Verify: (x - 2)(x - 3) = x² - 5x + 6 ✓
Example 2: Roots are 1/2 and -3
- Sum = 1/2 + (-3) = -5/2
- Product = (1/2)(-3) = -3/2
- Equation: x² - (-5/2)x + (-3/2) = 0
- x² + (5/2)x - 3/2 = 0
- Multiply by 2: 2x² + 5x - 3 = 0
Example 3: Roots are α and β where α + β = 5 and αβ = 4
- Directly use: x² - 5x + 4 = 0
- This factors as: (x - 1)(x - 4) = 0 → roots are 1 and 4
Key Points:
- Always express in standard form with integer coefficients (if possible)
- Multiply by LCM if fractions exist to clear denominators
- Verify by substituting the roots back
Exam Tip
Use the formula: x² - (sum)x + (product) = 0. Calculate sum and product carefully, especially with negative roots or fractions.
Common Mistake
Students write x² + (sum)x instead of x² - (sum)x. Remember: the coefficient of x is -1 times the sum. Also, clear fractions by multiplying by LCM.
Chapter Summary
Quadratic Equations are fundamental in algebra and have numerous real-world applications. This chapter covers:
- Standard Form: ax² + bx + c = 0 with a ≠ 0
- Factorization Method: Fast and effective when roots are rational
- Completing the Square: Universal method that works for all equations
- Quadratic Formula: Direct formula for any quadratic equation
- Nature of Roots: Using discriminant (Δ = b² - 4ac) to determine root type
- Vieta's Formulas: Relationships between roots and coefficients
- Forming Equations: Constructing equations from given roots
- Applications: Real-world problems involving quadratic equations
Exam Focus: All solving methods, discriminant analysis, nature of roots, Vieta's formulas, word problems.