Nature Of The Roots Of A Quadratic EquationThe nature of the roots depends on the value of b2 – 4ac. bx2 – 4ac is called the discriminant of the quadratic equation ax2 + bx + c = 0 and is generally, denoted by D. ∴ D = b2 – 4ac If D > 0, i..e., b2 – …

Relationship Between Zeros And Coefficients Of A PolynomialConsider quadratic polynomial P(x) = 2x2 – 16x + 30. Now, 2x2 – 16x + 30 = (2x – 6) (x – 3) = 2 (x – 3) (x – 5) The zeros of P(x) are 3 and 5. Sum of the zeros = 3 + 5 = 8 = \(\frac { -\left( -16…

Decimal Representation Of Rational NumbersExample 1:    Express \(\frac { 7 }{ 8 }\) in the decimal form by long division method. Solution:    We have, ∴ \(\frac { 7 }{ 8 }\) = 0.875Example 2:    Convert \(\frac { 35 }{ 16 }\) into decimal form by long…

What is a Ratio and Proportion RATIO In our day-to-day life, we compare one quantity with another quantity of the same kind by using the ‘method of subtraction’ and ‘method of division’. Example: The height of Seema is 1 m 67 cm and that of Reema is 1 m…

Factorization Of Polynomials Using Factor TheoremFactor Theorem:If p(x) is a polynomial of degree n  1 and a is any real number, then (i) x – a is a factor of p(x), if p(a) = 0, and (ii) p(a) = 0, if x – a is a factor of p(x). Proof: By the Remainder Th…

What Is The Cartesian Coordinate SystemIn Cartesian co-ordinates the position of a point P is determined by knowing the distances from two perpendicular lines passing through the fixed point. Let O be the fixed point called the origin and XOX’ and YOY’, …

Form A Polynomial With The Given ZerosLet zeros of a quadratic polynomial be α and β. x = β,               x = β x – α = 0,   x ­– β = 0 The obviously the quadratic polynomial is (x – α) (x – β) i.e.,  x2 – (α + β) x + αβ x2 – (Sum of the zeros)x + Prod…

What is the pattern of numbers?PATTERNS Patterns arise due to special arrangements of numbers associated with geometrical figures. The numbers are of different kinds likeNatural numbersPolygonal numbers (a) Triangular numbers (b) Square numbers (c) Penta…

Degree Of A PolynomialThe greatest power (exponent) of the terms of a polynomial is called degree of the polynomial. For example : In polynomial 5x2 – 8x7 + 3x: (i) The power of term 5x2 = 2 (ii) The power of term –8x7 = 7 (iii) The power of 3x = 1 Since…

Whole Numbers And Its PropertiesWHOLE NUMBERS Now if we add zero (0) in the set of natural numbers, we get a new set of numbers called the whole numbers. Hence the set of whole numbers consists of zero and the set of natural numbers. It is denoted by W. …