Topics
Number Systems
Algebra
Geometry
Trigonometry
Statistics and Probability
Coordinate Geometry
Mensuration
Internal Assessment
Real Numbers
Pair of Linear Equations in Two Variables
- Linear Equation in Two Variables
- Graphical Method of Solution of a Pair of Linear Equations
- Substitution Method
- Elimination Method
- Cross - Multiplication Method
- Equations Reducible to a Pair of Linear Equations in Two Variables
- Consistency of Pair of Linear Equations
- Inconsistency of Pair of Linear Equations
- Algebraic Conditions for Number of Solutions
- Simple Situational Problems
- Pair of Linear Equations in Two Variables
- Relation Between Co-efficient
Arithmetic Progressions
Quadratic Equations
- Quadratic Equations
- Solutions of Quadratic Equations by Factorization
- Solutions of Quadratic Equations by Completing the Square
- Nature of Roots of a Quadratic Equation
- Relationship Between Discriminant and Nature of Roots
- Situational Problems Based on Quadratic Equations Related to Day to Day Activities to Be Incorporated
- Application of Quadratic Equation
Polynomials
Circles
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Tangent to a Circle
- Number of Tangents from a Point on a Circle
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
Triangles
- Similar Figures
- Similarity of Triangles
- Basic Proportionality Theorem Or Thales Theorem
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Right-angled Triangles and Pythagoras Property
- Similarity Triangle Theorem
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Triangles Examples and Solutions
- Angle Bisector
- Similarity
- Ratio of Sides of Triangle
Constructions
Heights and Distances
Trigonometric Identities
Introduction to Trigonometry
Probability
Statistics
Lines (In Two-dimensions)
Areas Related to Circles
Surface Areas and Volumes
notes
In general, a real number α is called a root of the quadratic equation `ax^2 + bx + c = 0, a ≠ 0` if `aα^2 + bα + c = 0`. We also say that x = α is a solution of the quadratic equation, or that α satisfies the quadratic equation. Note that the zeroes of the quadratic polynomial `ax^2 + bx + c=0` and the roots of the quadratic equation `ax^2 + bx + c = 0` are the same.
By substituting arbitrary values for the variable and deciding the roots of quadratic equation is a time consuming process. Let us learn to use factorisation method to find the roots of the given quadratic equation.
`x^2 - 4 x - 5 = (x - 5) (x + 1)`
`(x - 5) and (x + 1)` are two linear factors of quadratic polynomial `x^2 - 4 x - 5.`
So the quadratic equation obtained from `x^2 - 4 x - 5` can be written as `(x - 5) (x + 1) = 0`
`x - 5 = 0 or x + 1 = 0`
∴`x = 5 or x = -1`
∴5 and the -1 are the roots of the given quadratic equation.
While solving the equation first we obtained the linear factors. So we call this method as ’factorization method’ of solving quadratic equation
Ex.(1) `m^2 - 14 m + 13 = 0`
∴`m^2 - 13 m - 1m + 13 = 0`
∴`m (m - 13) -1 (m - 13) = 0`
∴`(m - 13) (m - 1) = 0`
∴`m - 13 = 0 or m - 1 = 0`
∴`m = 13 or m = 1`
∴13 and 1 are the roots of the given quadratic equation.
