Quadratic Equations
A quadratic equation is a polynomial equation of degree two, meaning the highest power of the variable is 2. Quadratic equations form one of the most fundamental topics in Class 10 mathematics.
These equations appear in real-world problems involving projectile motion, area calculations, speed-distance-time relationships, and finding dimensions of objects.
Students must be able to recognise, formulate, and solve quadratic equations using four methods: factorisation, completing the square, the quadratic formula, and graphical method.
What is Quadratic Equations?
Definition: A quadratic equation is a second-degree polynomial equation in a single variable x, written in the standard form:
ax² + bx + c = 0, where a ≠ 0
Where:
- a, b, c are real numbers
- a ≠ 0 (otherwise it becomes a linear equation)
- x is the variable
The term "quadratic" comes from the Latin word quadratus (meaning square), because the variable is squared.
Terminology:
- ax² — quadratic term
- bx — linear term
- c — constant term
Roots of a quadratic equation:
- A quadratic equation has at most two roots (also called zeros or solutions).
- Roots are the values of x that make the equation equal to zero.
- Roots can be real and distinct, real and equal, or complex, depending on the discriminant D = b² − 4ac.
Quadratic equation vs. quadratic expression:
- ax² + bx + c is a quadratic polynomial (expression).
- ax² + bx + c = 0 is a quadratic equation.
Graphically, every quadratic equation corresponds to a parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward.
Quadratic Equations Formula
Standard Form:
ax² + bx + c = 0, where a ≠ 0
Quadratic Formula (Roots):
x = (−b ± √(b² − 4ac)) / 2a
Discriminant:
D = b² − 4ac
Sum and Product of Roots:
If α and β are the roots:
- Sum of roots: α + β = −b/a
- Product of roots: αβ = c/a
Derivation and Proof
The quadratic formula is derived by completing the square on the general equation.
Steps:
- Start with: ax² + bx + c = 0
- Divide by a (since a ≠ 0): x² + (b/a)x + c/a = 0
- Move constant to RHS: x² + (b/a)x = −c/a
- Add (b/2a)² to both sides: x² + (b/a)x + b²/(4a²) = −c/a + b²/(4a²)
- LHS is a perfect square: (x + b/(2a))² = (b² − 4ac) / (4a²)
- Take square root: x + b/(2a) = ± √(b² − 4ac) / (2a)
- Isolate x: x = (−b ± √(b² − 4ac)) / (2a)
The ± gives two solutions. The expression b² − 4ac under the square root is the discriminant, which determines the nature of the roots.
Types and Properties
Types by Structure:
| Type | Description | Example |
|---|---|---|
| Complete Quadratic | All three terms present (b ≠ 0, c ≠ 0) | 2x² + 5x + 3 = 0 |
| Pure Quadratic | Only quadratic and constant terms (b = 0) | 3x² − 27 = 0 |
| Incomplete (c = 0) | Constant term missing; factorable with x as common factor | 4x² − 12x = 0 |
Types by Nature of Roots:
| Discriminant (D = b² − 4ac) | Nature of Roots | Graph |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola cuts x-axis at two points |
| D = 0 | Two equal real roots (repeated) | Parabola touches x-axis at one point |
| D < 0 | No real roots (complex) | Parabola does not intersect x-axis |
Methods
There are four methods for solving quadratic equations:
Method 1: Factorisation (Splitting the Middle Term)
- Express the middle term bx as the sum of two terms whose product equals ac.
- Factor by grouping into two linear factors.
Example: x² + 5x + 6 = 0
- Find two numbers that add to 5 and multiply to 6: 2 and 3.
- x² + 2x + 3x + 6 = 0 → x(x + 2) + 3(x + 2) = 0 → (x + 2)(x + 3) = 0
- x = −2 or x = −3
Method 2: Completing the Square
Transform the equation into the form (x + p)² = q, then take square roots. This method works for all quadratic equations and is the basis for deriving the quadratic formula.
Method 3: Quadratic Formula
Substitute a, b, c into x = (−b ± √(b² − 4ac)) / 2a. This method always works, regardless of whether roots are rational, irrational, or complex.
Method 4: Graphical Method
Plot y = ax² + bx + c. The x-intercepts of the parabola give the roots.
Comparison:
| Method | Best Used When | Limitation |
|---|---|---|
| Factorisation | Roots are integers or simple fractions | Fails for irrational or complex roots |
| Completing the Square | Deriving the formula; coefficient of x² is 1 | Tedious for large coefficients |
| Quadratic Formula | Always applicable | Requires careful arithmetic |
| Graphical Method | Visual understanding | Less precise; impractical in exams |
Solved Examples
Example 1: Solving by Factorisation (Simple)
Problem: Solve x² − 7x + 12 = 0 by factorisation.
Solution:
Given:
- a = 1, b = −7, c = 12
Using Factorisation:
- Find two numbers that multiply to 12 and add to −7: −3 and −4.
- Split: x² − 3x − 4x + 12 = 0
- Group: x(x − 3) − 4(x − 3) = 0
- Factor: (x − 3)(x − 4) = 0
- x − 3 = 0 or x − 4 = 0
Answer: x = 3 or x = 4
Example 2: Solving a Pure Quadratic Equation
Problem: Solve 5x² − 80 = 0.
Solution:
Given:
- a = 5, b = 0, c = −80 (pure quadratic)
Steps:
- 5x² = 80
- x² = 16
- x = ± √16 = ± 4
Answer: x = 4 or x = −4
Example 3: Solving When Constant Term is Zero
Problem: Solve 3x² − 15x = 0.
Solution:
Given:
- a = 3, b = −15, c = 0
Steps:
- Take x as common factor: x(3x − 15) = 0
- x = 0 or 3x − 15 = 0
- 3x = 15, so x = 5
Answer: x = 0 or x = 5
Example 4: Factorisation with Leading Coefficient Greater Than 1
Problem: Solve 6x² + x − 15 = 0 by factorisation.
Solution:
Given:
- a = 6, b = 1, c = −15
- Product ac = 6 × (−15) = −90
Using Factorisation:
- Find two numbers that multiply to −90 and add to 1: 10 and −9.
- Split: 6x² + 10x − 9x − 15 = 0
- Group: 2x(3x + 5) − 3(3x + 5) = 0
- Factor: (3x + 5)(2x − 3) = 0
- 3x + 5 = 0 → x = −5/3
- 2x − 3 = 0 → x = 3/2
Answer: x = −5/3 or x = 3/2
Example 5: Solving by Completing the Square
Problem: Solve x² + 6x + 2 = 0 by completing the square.
Solution:
Given:
- a = 1, b = 6, c = 2
Steps:
- Move constant: x² + 6x = −2
- Half of coefficient of x: 6/2 = 3. Square it: 3² = 9.
- Add 9 to both sides: x² + 6x + 9 = −2 + 9 = 7
- Perfect square: (x + 3)² = 7
- Take square root: x + 3 = ± √7
- x = −3 ± √7
Answer: x = −3 + √7 or x = −3 − √7 (approximately −0.354 or −5.646)
Example 6: Using the Quadratic Formula
Problem: Solve 2x² − 4x − 3 = 0 using the quadratic formula.
Solution:
Given:
- a = 2, b = −4, c = −3
Using x = (−b ± √D) / 2a:
- D = (−4)² − 4(2)(−3) = 16 + 24 = 40
- x = (−(−4) ± √40) / (2 × 2) = (4 ± √40) / 4
- Simplify √40 = 2√10: x = (4 ± 2√10) / 4 = (2 ± √10) / 2
Answer: x = (2 + √10)/2 or x = (2 − √10)/2 (approximately 2.581 or −0.581)
Example 7: Forming a Quadratic Equation from Roots
Problem: Form a quadratic equation whose roots are 5 and −2.
Solution:
Given:
- α = 5, β = −2
Using Sum and Product of Roots:
- Sum: α + β = 5 + (−2) = 3
- Product: αβ = 5 × (−2) = −10
Quadratic equation: x² − (sum)x + (product) = 0
Answer: x² − 3x − 10 = 0
Example 8: Word Problem: Finding Consecutive Numbers
Problem: The product of two consecutive positive integers is 182. Find the integers.
Solution:
Given:
- Product of two consecutive positive integers = 182
Let the integers be x and x + 1.
Steps:
- x(x + 1) = 182
- x² + x − 182 = 0
- Find two numbers that multiply to −182 and add to 1: 14 and −13.
- Factorise: (x + 14)(x − 13) = 0
- x = −14 or x = 13. Since integers are positive, x = 13.
Answer: The integers are 13 and 14.
Example 9: Word Problem: Speed and Distance
Problem: A train travels 480 km at uniform speed. If the speed were 8 km/h less, the train would take 3 hours more. Find the speed.
Solution:
Given:
- Distance = 480 km
- Reduced speed = (x − 8) km/h takes 3 hours more
Let speed = x km/h. Time = 480/x hours.
Steps:
- Condition: 480/(x − 8) − 480/x = 3
- Multiply by x(x − 8): 480x − 480(x − 8) = 3x(x − 8)
- Simplify: 3840 = 3x² − 24x
- 3x² − 24x − 3840 = 0. Divide by 3: x² − 8x − 1280 = 0
- Factorise: (x − 40)(x + 32) = 0
- x = 40 or x = −32. Speed cannot be negative.
Answer: The speed is 40 km/h.
Example 10: Word Problem: Area of a Rectangle
Problem: The length of a garden is 3 m more than twice its breadth. The area is 170 m². Find the dimensions.
Solution:
Given:
- Length = 2x + 3 (where x = breadth)
- Area = 170 m²
Steps:
- x(2x + 3) = 170
- 2x² + 3x − 170 = 0
- Using quadratic formula: a = 2, b = 3, c = −170
- D = 9 + 1360 = 1369 = 37²
- x = (−3 ± 37)/4
- x = 34/4 = 8.5 or x = −40/4 = −10
- Breadth cannot be negative, so x = 8.5 m.
- Length = 2(8.5) + 3 = 20 m.
Answer: Breadth = 8.5 m, Length = 20 m.
Real-World Applications
Physics and Engineering:
- Projectile motion follows a parabolic path described by a quadratic equation.
- Engineers calculate rocket trajectories, bridge shapes, and satellite dish curvature.
Architecture and Construction:
- Designing arches, domes, and parabolic structures requires quadratic equations.
- Area calculations for rooms, gardens, and plots often lead to quadratic equations.
Business and Economics:
- Profit maximisation and cost minimisation use quadratic functions.
- Revenue is modelled as a quadratic function of units sold.
- Break-even analysis requires solving quadratic equations.
Sports:
- Trajectory of a ball in cricket, football, or basketball follows a quadratic model.
- Used to predict outcomes and improve performance.
Daily Life:
- Problems involving age, distribution of items, speed/travel time, and dimensions.
- Calculating time for a dropped object to hit the ground is a classic quadratic problem.
Key Points to Remember
- A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0.
- A quadratic equation has at most two roots (real or complex).
- Four methods of solving: factorisation, completing the square, quadratic formula, graphical method.
- Discriminant D = b² − 4ac determines the nature of roots: D > 0 (two distinct real), D = 0 (two equal real), D < 0 (no real).
- Sum of roots = −b/a; product of roots = c/a.
- A quadratic equation can be formed from roots: x² − (sum)x + (product) = 0.
- Always check the equation is in standard form before applying any method.
- In word problems, verify solutions make sense (reject negative values for length, speed, or age).
- Factorisation is quickest for rational roots; use the quadratic formula for all other cases.
- Every quadratic equation represents a parabola on a graph; roots are the x-intercepts.
Practice Problems
- Solve by factorisation: x² − 9x + 20 = 0
- Solve by factorisation: 2x² + 7x − 15 = 0
- Solve by completing the square: x² − 8x + 5 = 0
- Using the quadratic formula, solve: 3x² + 11x + 6 = 0
- The sum of a number and its reciprocal is 26/5. Find the number.
- A rectangular field has a perimeter of 82 metres and an area of 400 square metres. Find the dimensions of the field.
- The product of two consecutive even numbers is 528. Find the numbers.
- A motor boat whose speed in still water is 18 km/h takes 1 hour more to go 24 km upstream than to return downstream. Find the speed of the stream.
Frequently Asked Questions
Q1. What is a quadratic equation in maths?
A quadratic equation is a polynomial equation of degree 2 in the standard form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. The variable is raised to the power of 2 as its highest exponent.
Q2. How many roots can a quadratic equation have?
A quadratic equation can have at most two roots. These can be two distinct real numbers, two equal real numbers (a repeated root), or two complex conjugate numbers. The nature depends on the discriminant.
Q3. What is the difference between a quadratic equation and a quadratic expression?
A quadratic expression is a polynomial of degree 2, such as ax² + bx + c. A quadratic equation is formed when this expression is set equal to zero: ax² + bx + c = 0. The equation asks for the values of x that make the expression zero.
Q4. Why is a ≠ 0 required in a quadratic equation?
If a = 0, the x² term disappears and the equation becomes bx + c = 0, which is linear (degree 1), not quadratic (degree 2). The condition a ≠ 0 ensures the equation remains second-degree.
Q5. Which method is best for solving quadratic equations?
Factorisation is quickest when the equation has simple, rational roots. When factorisation is difficult, the quadratic formula is the most reliable method — it works for all quadratic equations regardless of the nature of roots.
Q6. Can a quadratic equation have only one root?
When D = 0, the equation has two equal roots (same value). Technically it has two roots that happen to be identical. This is described as a 'repeated root' or 'double root'.
Q7. What is the relationship between roots and coefficients?
For ax² + bx + c = 0 with roots α and β: sum of roots = −b/a, product of roots = c/a. This relationship is used for forming equations and verifying solutions.
Q8. How do quadratic equations appear in CBSE Class 10 board exams?
Quadratic equations typically carry 6–10 marks. Questions include solving by factorisation, using the quadratic formula, finding the nature of roots using the discriminant, and solving word problems involving areas, speeds, and ages.
Q9. Is x² = 25 a quadratic equation?
Yes. It can be rewritten as x² − 25 = 0 (standard form with a = 1, b = 0, c = −25). It is a pure quadratic equation since the linear term is missing.
Q10. What are some real-life examples of quadratic equations?
Calculating the path of a thrown ball (projectile motion), determining field area with perimeter constraints, computing business profit, and finding object dimensions when length-breadth relationships are given.
Related Topics
- Standard Form of Quadratic Equation
- Solving Quadratic Equations by Factorisation
- Quadratic Formula
- Discriminant of Quadratic Equation
- Factorisation of Quadratic Expressions
- Completing the Square Method
- Nature of Roots of Quadratic Equation
- Sum and Product of Roots
- Word Problems on Quadratic Equations
- Applications of Quadratic Equations
- Roots and Graphs of Quadratic Equations
