#### Topics

##### Number Systems

##### Real Numbers

##### Algebra

##### 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

##### Geometry

##### 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 (Thales Theorem)
- Criteria for Similarity of Triangles
- Areas of Similar Triangles
- Right-angled Triangles and Pythagoras Property
- Similarity of Triangles
- Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
- Triangles Examples and Solutions
- Angle Bisector
- Similarity of Triangles
- Ratio of Sides of Triangle

##### Constructions

##### Trigonometry

##### Heights and Distances

##### Trigonometric Identities

##### Introduction to Trigonometry

##### Statistics and Probability

##### Probability

##### Statistics

##### Coordinate Geometry

##### Lines (In Two-dimensions)

##### Mensuration

##### Areas Related to Circles

##### Surface Areas and Volumes

- Concept of Surface Area, Volume, and Capacity
- Surface Area of a Combination of Solids
- Volume of a Combination of Solids
- Conversion of Solid from One Shape to Another
- Frustum of a Cone
- Concept of Surface Area, Volume, and Capacity
- Surface Area and Volume of Different Combination of Solid Figures
- Surface Area and Volume of Three Dimensional Figures

##### Internal Assessment

## Notes

have seen that the roots of the equation ax^{2} + bx + c = 0 are given by

`x=(-b+-sqrt(b^2-4ac))/(2a)`

If b^{2} – 4ac > 0, we get two distinct real roots, `-b/(2a)+(b^2-4ac)/(2a)` and `-b/(2a)-(sqrt(b^2-4ac))/(2a)`

If b^{2} – 4ac = 0,then x= `-b/(2a)+-0` i.e., `x=-b/(2a) or -b/(2a)`

So, the roots of the equation ax^{2} + bx + c = 0 are both `-b/(2a)`

Therefore, we say that the quadratic equation ax^{2} + bx + c = 0 has two equal real roots in this case.

If b^{2} – 4ac < 0, then there is no real number whose square is b^{2} – 4ac. Therefore, there are no real roots for the given quadratic equation in this case.

Since b^{2} – 4ac determines whether the quadratic equation ax^{2} + bx + c = 0 has real roots or not, b2 – 4ac is called the **discriminant** of this quadratic equation.

So, if a quadratic equation ax^{2} + bx + c = 0 has

(i) two distinct real roots, if b^{2} – 4ac > 0,

(ii) two equal real roots, if b^{2} – 4ac = 0,

(iii) no real roots, if b^{2} – 4ac < 0.

Let us consider one examples.

Example : Find the discriminant of the quadratic equation 2x^{2} – 4x + 3 = 0, and hence find the nature of its roots.

Solution : The given equation is of the form ax^{2} + bx + c = 0, where a = 2, b = – 4 and c = 3.

Therefore, the discriminant

b^{2} – 4ac = (– 4)^{2} – (4 × 2 × 3) = 16 – 24 = – 8 < 0

So, the given equation has no real roots.