10th Maths Chapter 3 Algebra

Key formulas

• Roots of a quadratic equation

  $ax^2 + bx + c = 0,\; (a \neq 0) \;\Rightarrow\; x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Sum of the roots

  $\alpha + \beta = \dfrac{-b}{a} = \dfrac{-\text{Co-efficient of } x}{\text{Co-efficient of } x^2}$

• Product of the roots

  $\alpha\beta = \dfrac{c}{a} = \dfrac{\text{Constant term}}{\text{Co-efficient of } x^2}$

• Quadratic equation formed from its roots

  $x^2 - (\alpha + \beta)x + \alpha\beta = 0$

• Discriminant

  $\Delta = b^2 - 4ac$

• Nature of roots: real and unequal

  $\Delta > 0 \;\Rightarrow\; \text{the roots are real and unequal}$

• Nature of roots: real and equal

  $\Delta = 0 \;\Rightarrow\; \text{the roots are real and equal}$

• Nature of roots: no real roots

  $\Delta < 0 \;\Rightarrow\; \text{there are no real roots}$

• Cases for a system of linear equations in three variables

  $\text{(i) Unique solution}\quad \text{(ii) Infinitely many solutions}\quad \text{(iii) No solution}$

• LCM of algebraic expressions

  $\text{LCM} = \text{expression of lowest degree that each given expression exactly divides}$

• Quadratic polynomial and its zeroes

  $p(x) = ax^2 + bx + c \;(a \neq 0) \;\Rightarrow\; p(x) \text{ has atmost two zeroes, which lie on the } x\text{-axis}$

• Order of a matrix

  $\text{Order of } A = (\text{Number of rows}) \times (\text{Number of columns}) = m \times n$

• Row matrix (row vector)

  $A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \end{pmatrix}_{1 \times n}$

• Column matrix (column vector)

  $A = \begin{pmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{pmatrix}_{m \times 1}$

• Square matrix

  $\text{Number of rows} = \text{Number of columns},\; \text{order } n \times n$

• Zero matrix (null matrix)

  $a_{ij} = 0 \text{ for all } i,\, j$

• Transpose of a matrix

  $\left(A^{T}\right)_{ij} = a_{ji} \quad (\text{rows and columns of } A \text{ interchanged})$

• Diagonal matrix

  $a_{ij} = 0 \text{ for } i \neq j \quad (\text{square matrix})$

• Scalar matrix

  $a_{ij} = \begin{cases} k, & i = j \\ 0, & i \neq j \end{cases}$

• Identity matrix (unit matrix)

  $a_{ij} = \begin{cases} 1, & i = j \\ 0, & i \neq j \end{cases}$

• Lower triangular matrix

  $a_{ij} = 0 \text{ for } i < j \quad (\text{all entries above the leading diagonal are zero})$

• Upper triangular matrix

  $a_{ij} = 0 \text{ for } i > j \quad (\text{all entries below the leading diagonal are zero})$

• Equality of matrices

  $A = B \iff \text{same order and } a_{ij} = b_{ij} \text{ for all } i,\, j$