PUMPA - SMART LEARNING
எங்கள் ஆசிரியர்களுடன் 1-ஆன்-1 ஆலோசனை நேரத்தைப் பெறுங்கள். டாப்பர் ஆவதற்கு நாங்கள் பயிற்சி அளிப்போம்
Book Free DemoIdentities are the algebraic equation which is always true regardless of the values assigned to the variables.
The identities give an alternative method of solving problems on multiplication of algebraic expressions and also of numbers.
Let us recall some of the squaring identities.
Let us derive the formula by applying distributive property \(a(b+c)\) \(=\) \(ab+bc\).
1. We will prove \((a+b)^2\) \(=\) \(a^2+2ab+b^2\).
Consider the LHS \((a+b)^2\).
We know that the square number/term/expression is the product of itself.
Thus, \((a+b)^2 =\) \((a+b)(a+b)\).
Apply the distributive property.
\((a+b)(a+b)\) \(=\) \((a\times a)\)\(+(a\times b)\)\(+(b\times a)\)\(+(b\times b)\)
\(= a^2+ab+ba+b^2\)
\(=a^2+ab+ab+b^2\)
\(=a^2+2ab+b^2\) \(=\) RHS
So \((a+b)^2\) \(=\) \(a^2+2ab+b^2\).
2. We will prove \((a-b)^2\) \(=\) \(a^2-2ab+b^2\).
Consider the LHS \((a-b)^2\).
We know that the square number/term/expression is the product of itself.
Thus, \((a-b)^2\) \(=\) \((a-b)(a-b)\).
Apply the distributive property.
\((a-b)(a-b)\) \(=\) \((a\times a)\)\(+(a\times -b)\)\(+(-b\times a)\)\(+(-b\times -b)\)
\(= a^2-ab-ba+b^2\)
\(=a^2-ab-ab+b^2\)
\(=a^2-2ab+b^2\) \(=\) RHS
So \((a-b)^2\) \(=\) \(a^2-2ab+b^2\).
3. We will prove \((a+b)(a-b)\) \(=\) \(a^2-b^2\).
Consider the LHS \((a+b)(a-b)\).
Apply distributive property.
\((a+b)(a-b)\) \(=\) \((a\times a)\)\(+(a\times -b)\)\(+(b\times a)\)\(+(b\times -b)\)
\(= a^2-ab+ba-b^2\)
\(= a^2-b^2\) \(=\) RHS
So \((a+b)(a-b)\) \(=\) \(a^2-b^2\).
The identities \(1\), \(2\) and \(3\) are known as Standard identities.
4. We will now prove \((x+a)(x+b)\) \(=\) \(x^2\)\(+(a+b)x\)\(+ab\).
Consider the LHS \((x+a)(x+b)\).
Apply the distributive property.
\((x+a)(x+b)\) \(=\) \((x\times x)\)\(+(x\times b)\)\(+(a\times x)\)\(+(a\times b)\)
\(= x^2+xb+ax+ab\)
\(= x^2+ax+bx+ab\)
\(= x^2+(a+b)x+ab\) \(=\) RHS
So \((x+a)(x+b)\) \(=\) \(x^2\)\(+(a+b)x\)\(+ab\).
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