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4. Algebraic Expressions and Algebraic Formulas eLearn.Punjab

Example
If a + b = 7and a – b = 3, then find the value of (a) a2 + b2 (b) ab

Solution

We are given that a + b = 7 and a - b = 3

(a) To find the value of (a2 + b2), we use the formula

(a + b)2 + (a - b)2 = 2(a2 + b2)

Substituting the values a + b = 7 and a - b = 3, we get

(7)2 + (3)2 = 2(a2 + b2)

⇒ 49 + 9 = 2(a2 + b2)

⇒ 58 = 2(a2 + b2) ....(simplifying)

⇒ 29 = a2 + b2 ....(dividing by 2)

(b) To find the value of ab, we make use of the formula

(a + b)2 - (a - b)2 = 4ab

(7)2 - (3)2 = 4ab

⇒ 49 - 9 = 4ab

⇒ 40 = 4ab ....(simplifying)

⇒ 10 = ab ....(dividing by 4)

Hence a2 + b2 = 29 and ab = 10.

Version: 1.1 (ii) (a + b + c)2 = a 2 + b2 + c2 + 2ab + 2bc + 2ca
This formula, square of a trinomial, involves three expressions,

namely; (a + b + c), (a2 + b2 + c2) and 2(ab + bc + ca). If the values
of two of them are known, the value of the third expression can be
calculated. The method is explained in the following examples.

Example 1
If a2 + b2 + c2 = 43 and ab + bc + ca = 3, then find the value of

a + b + c.

Solution
We know that
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

⇒ (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

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