Question

find the value using suitable property a)97² b)101×96​

Answer 1

$\Large{\underbrace{\sf{\purple{Required\:Solution:}}}}$

(a) (97)²

Solution - $\displaystyle{\sf{ (100-3)^{2}}}$

• Using Identity — (α-b)² = α²- 2αb + b²

$\longrightarrow{\sf{ (100)^{2} - 2\times 100\times 3 + (3)^{2}}}$

$\longrightarrow{\sf{ 10,000 - 600 + 9}}$

$\longrightarrow{\sf{ 9,400 + 9}}$

$\large\implies{\boxed{\sf{\purple{ 9,409}}}}$

(b) 101 × 96

Solution - $\displaystyle{\sf{ (100+1)(100-4)}}$

• Using Identity - (x+α)(x+b) = x²+(α+b)x + αb

$\longrightarrow{\sf{ (100)^{2} + \big[1+(-4)\big]\times 100 + \big[1\times (-4)\big]}}$

$\longrightarrow{\sf{ 10,000 + (-3)\times 100 +(-4)}}$

$\longrightarrow{\sf{ 10,000 + (-300) +(-4)}}$

$\longrightarrow{\sf{ 10,000 -304}}$

$\large\implies{\boxed{\sf{\purple{9,696}}}}$

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$\Large{\underbrace{\sf{\purple{Explore\:More!}}}}$

Algebrαic identity - An αlgebrαic identity is αn equαlity thαt holds for αny vαlues of its vαriαbles.

e.g.,

• (a+b)² = a² + 2ab + b²
• (a-b)² = a² - 2ab + b²
• a²-b² = (a-b)(a+b)
• a²+b² = (a+b)² - 2ab
• (x+a)(x+b) = x² + (a+b)x + ab
• (a+b)³ = a³ + b³ + 3ab(a+b)
• (a-b)³ = a³ - b³ - 3ab(a-b)
• a³+b³ = (a+b)³-3ab(a+b)

⠀⠀⠀⠀⠀⠀⠀= (a+b) (a²- ab + b²)

• a³-b³ = (a-b)³+3ab(a-b)

⠀⠀⠀⠀⠀⠀⠀= (a-b) (a² + ab + b²)

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

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Answer 2

$\bf\huge{Solution:}$

A) (97)²

$\to$ $\displaystyle{\bf{ (100-3)^{2}}}$

Identity:- (a-b)² = a²- 2ab + b²

$\implies{\rm{ (100)^{2} - 2\times 100\times 3 + (3)^{2}}}$

$\implies{\rm{ 10,000 - 600 + 9}}$

$\implies{\rm{ 9,400 + 9}}$

$\to\rm\large{9,409}$

__________________________________________

B) 101 × 96

$\displaystyle{\bf{ (100+1)(100-4)}}$

Identity:- (x+a)(x+b) = x²+(a+b)x + ab

$\implies{\rm{ (100)^{2} + \big[1+(-4)\big]\times 100 + \big[1\times (-4)\big]}}$

$\implies{\rm{ 10,000 + (-3)\times 100 +(-4)}}$

$\implies{\rm{ 10,000 + (-300) +(-4)}}$

$\implies{\rm{ 10,000 -304}}$

$\to\rm\large{9,696}$

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