# Basic Formulae

Algebra requires dealing with unknown quantities. We use symbols to get general results. Hence it is useful to know the basic formulae.

I. (a + b)^{2} = a^{2} + b^{2} + 2ab.

II. (a â€“ b)^{2} = a^{2} + b^{2 }â€“ 2ab.

III. (a^{2 }â€“ b^{2}) = (a + b) (a â€“ b).

IV. (a + b)^{2} - (a-b)^{2} = 4ab.

V. (a + b)^{2} + (a-b)^{2} = 2(a^{2 }+ b^{2}).

VI. (a + b)^{3} = a^{3} + b^{3} + 3ab(a + b).

VII. (a â€“ b)^{3} = a^{3} - b^{3} - 3ab(a â€“ b).

VIII. (a^{3 }+ b^{3}) = (a + b)(a^{2 }+ b^{2 }â€“ ab).

IX. (a^{3 }â€“ b^{3}) = (a â€“ b) (a^{2 }+ b^{2 }+ ab).

X. (a + b + c)^{2} = [a^{2 }+ b^{2 }+ c^{2 }+ 2(ab + bc + ca)].

XI. (a + b + c + d)^{2} = [a^{2 }+ b^{2 }+ c^{2 }+ d^{2}+ 2a(b + c + d) + 2b(c + d) + 2cd].

XII. (a^{3 }+ b^{3 }+ c^{3 }â€“3abc) = (a + b + c) (a^{2 }+ b^{2 }+ c^{2} â€“ ab â€“ bc â€“ ca). If a + b + c = 0 â‡’ a^{3 }+ b^{3 }+ c^{3} = 3abc.

XIII. (x + a)(x + b) = x^{2}+ (a + b)x + ab.

XIV. (x + a)(x + b)(x + c) = x^{3 }+ (a + b + c) x^{2} + (ab + bc + ca) x + abc.

# Illustrations

**Illustration 1**:

If x + 1/x = 7, what is the value of x^{2} + 1/x^{2}?

x + 1/x = 7. Squaring both sides, we get x^{2} + 1/x^{2} + 2x(1/x) = 49;

hence x^{2} + 1/x^{2} = 49 â€“ 2 = 47.

**Illustration 2**:

Given x^{2} + 1/x^{2 }= 27, find the value of x â€“ 1/x.

(x â€“ 1/x)^{2} = x^{2} + 1/x^{2} â€“ 2x(1/x) = x^{2} + 1/x^{2} â€“ 2.

Hence x^{2} + 1/x^{2} = 27 â€“ 2; (x â€“ 1/x)^{2} = 25 â‡’ x â€“ 1/x = âˆš25 = Â± 5.

**Illustration 3**:

Find the value of following expression:

__6.79 ____Ã—____ 6.79 ____Ã—____ 6.79 + 3.21 ____Ã—____ 3.21 ____Ã—____ 3.21__

6.79 Ã— 6.79 â€“ 6.79 Ã— 3.21 + 3.21 Ã— 3.21

The expression is of the form (a^{3} â€“ b^{3})/(a^{2} â€“ ab + b^{2}). By using the above formulae, we can cancel out the terms leaving (a + b). Substitute â€˜aâ€™ = 6.79 and â€˜bâ€™ = 3.21, hence a + b = 6.79 + 3.21 = 10.

# Basics of Algebra

**Forming equations**. One problem in algebra is how to formulate equations.

Letters such as x or n are used to represent unknown quantities. For example, if Anil has 5 more pencils than Bela, then, if x represents the number of pencils that Bela has, then the number of pencils that Anil has is x + 5.

Or, if Subhashâ€™s present salary y is increased by 7%, then his new salary is 1.07y.

A combination of letters and arithmetic operations, such as x + 5, 3x^{2}/ (2x â€“ 5) and 19x^{2 }- 6x + 3, are called algebraic expressions.

**Coefficients**. The expression 19x^{2 }- 6x + 3 consists of the terms 19x^{2}, -6x, and 3, where 19 is the coefficient of x^{2}, - 6 is the coefficient of x, and 3 is a constant term (or coefficient of x^{0} = 1). Such an expression is called a second degree (or quadratic) polynomial in x since the highest power of x is 2. The expression x + 5 is a first degree (or linear) polynomial in x since the highest power of variable x is 1.

**Simplification.** Algebraic expressions can be simplified by factoring or combining like terms. For example, 6x + 5x is equivalent to (6 + 5)x. The expression 9x - 3y can be written as = 3(3x - y). In the expression 5x^{2} + 6y, there are no like terms and no common factors.

**Multiplication.** To multiply two algebraic expressions, each term of one expression is multiplied by each term of the other expression.

For example: (3x - 4) (9y + x) = 3x (9y + x) â€“ 4 (9y + x) =

(3x) (9y) + (3x) (x) + (-4) (9y) + (-4) (x) = 27xy + 3x^{2} - 36y - 4x.