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Let us expand some of the squared terms using the suitable standard identities.
1. \((2x+3y)^2\).
Let us use the identity, \((a+b)^2\) \(=\) \(a^2+2ab+b^2\).
Comparing \((2x+3y)^2\) with \((a+b)^2\), we have \(a=2x\) and \(b=3y\).
Substitute the values in the formula.
\((2x+3y)^2\) \(=\) \((2x)^2+(3y)^2+2(2x)(3y)\)
\((2x+3y)^2\) \(=\) \(4x^2+9y^2+12xy\).
2. \((5x-7y)^2\).
Let us use the identity, \((a-b)^2\) \(=\) \(a^2-2ab+b^2\).
Comparing \((5x-7y)^2\) with \((a-b)^2\), we have \(a=5x\) and \(b=7y\).
Substitute the values in the formula.
\((5-7y)^2\) \(=\) \((5x)^2+ (7y)^2-2(5x)(7y)\)
\((5-7y)^2\) \(=\) \(25x^2+49y^2-70xy\).
3. \((x+5y)(x-5y)\).
Let us use the identity, \((a+b)(a-b)\) \(=\) \(a^2-b^2\).
Comparing \((x+5y)(x-5y)\) with \((a+b)(a-b)\), we have \(a=x\) and \(b=5y\).
Substitute the values in the formula.
\((x+5y)(x-5y)\) \(=\)\((x)^2-(5y)^2\)
\((x+5y)(x-5y)\) \(=\)\(x^2-25y^2\).
4. \((4y+5)(4y+3)\).
Let us use the identity, \((x+a)(x+b)\) \(=\) \(x^2+(a+b)x+ab\).
Comparing \((4y+5)(4y+3)\) with \((x+a)(x+b)\), we have \(x=4y\), \(a=5\) and \(b=3\).
Substitute the values in the formula.
\((4y+5)(4y+3)\) \(=\) \((4y)^2+(5+3)(4y)+(5)(3)\)
\((4y+5)(4y+3)\) \(=\) \(16y^2+32y+15\).
Example:
Look for the following cases where we used the identities.
1. Expand \((x+4)^2\) using the identity.
The above expression is in \((a+b)^2\) form.
We have the identity, \((a+b)^2\) \(=\) \(a^2+2ab+b^2\).
Substitute \(a = x\) and \(b = 4\) in the formula.
\((x+4)^2\) \(=\) \(x^2+2(x)(4)+4^2\)
\(=\) \(x^2+2\times 4x+16\)
\(=\) \(x^2+8x+16\)
2. Evaluate \(98^2\) using identity.
\(98^2\) \(=\) \((100-2)^2\)
The above expression is in \((a-b)^2\) form.
We have the identity, \((a-b)^2\) \(=\) \(a^2-2ab+b^2\).
Substitute \(a = 100\) and \(b = 2\) in the formula.
\((100-2)^2\) \(=\) \(100^2-2(100)(2)+2^2\)
\(= 10000-400+4\)
\(= 9604\)
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