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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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