Question

Find each of the following products applying identities

a ) (3x + 5y) (3x + 5y)

b ) (3x² + 2) (3x² + 2)

$\sf c) \: ( \dfrac{2}{3} x + \dfrac{3}{4} y)( \dfrac{2}{3} x + \dfrac{3}{4} y)$
$\sf d) \: ( \dfrac{3}{4} {x}^{2} + 5) ( \dfrac{3}{4} {x}^{2} + 5)$

​

Answer 1

$\begin{gathered}\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept\;:-}}}\end{gathered}$

here we have to use the concept of algebraic identities and we have to multiply the expression by themselves only .

∴ we can apply identities to solve them .

_______________________________________________

IDENTITIES WHICH WE HAVE TO USE TO SOLVE THESE EQUATIONS :-

$\begin{gathered}\\\;\underline{\boxed{\tt{( a + b ) ^{2} = a ^{2} + 2ab + b ^{2} }}}\end{gathered}$

_______________________________________________

† ANSWER :-

a ) 9x² + 30xy + 25y²

b ) 9x⁴ + 12x² + 4

$\sf c) \: \dfrac{4}{9} {x}^{2} + xy + \dfrac{9}{16} {y}^{2}$

$\sf d) \: \dfrac{9}{16} {x}^{4} + \dfrac{15}{2} {x}^{2} + 25$

_______________________________________________

a ) (3x + 5y) (3x + 5y)

$\begin{gathered}:\Longrightarrow \sf(3x + 5y) ^{2} \end{gathered}$

By applying the identity

★ [ (a+b)² = a² + 2ab + b² ]

$\begin{gathered}:\Longrightarrow \sf(3x) ^{2} +2 \times 3x \times 5y + ( 5y) ^{2} \end{gathered}$

$\begin{gathered}:\Longrightarrow \sf {9x}^{2} + 30xy + {25y}^{2} \end{gathered}$

_______________________________________________

b ) (3x² + 2) (3x² + 2)

$\begin{gathered}:\Longrightarrow \sf(3x ^{2} + 2) ^{2} \end{gathered}$

By applying the identity

★ [ (a+b)² = a² + 2ab + b² ]

$\begin{gathered}:\Longrightarrow \sf(3x ^{2} )^{2} + 2 \times {3x}^{2} \times 2 + (2) ^{2} \end{gathered}$

$\begin{gathered}:\Longrightarrow \sf {9x}^{4} + {12x}^{2} + 4 \end{gathered}$

_______________________________________________

$\sf c) \: ( \dfrac{2}{3} x + \dfrac{3}{4} y)( \dfrac{2}{3} x + \dfrac{3}{4} y)$

$\begin{gathered}:\Longrightarrow \sf ( \dfrac{2}{3} x + \dfrac{3}{4} y) ^{2} \end{gathered}$

By applying the identity

★ [ (a+b)² = a² + 2ab + b² ]

$\begin{gathered}:\Longrightarrow \sf ( \dfrac{2}{3} x )^{2} + 2 \times \dfrac{2}{3} x \times \dfrac{3}{4} y + (\dfrac{3}{4} y) ^{2} \end{gathered}$

$\begin{gathered}:\Longrightarrow \sf \frac{4}{9} {x}^{2} + xy + \frac{9}{16} {y}^{2} \end{gathered}$

_______________________________________________

$\sf d) \: ( \dfrac{3}{4} {x}^{2} + 5) ( \dfrac{3}{4} {x}^{2} + 5)$

$\begin{gathered}:\Longrightarrow \sf ( \dfrac{3}{4} x ^{2} + 5) ^{2} \end{gathered}$

By applying the identity

★ [ (a+b)² = a² + 2ab + b² ]

$\begin{gathered}:\Longrightarrow \sf ( \dfrac{3}{4} x ^{2} )^{2} + 2 \times \dfrac{3}{4} x ^{2}\times 5 + (5) ^{2} \end{gathered}$

$\begin{gathered}:\Longrightarrow \sf \frac{9}{16} {x}^{4} + \frac{15}{2} {x}^{2} + 25 \end{gathered}$

_______________________________________________

$\begin{gathered}\\\;\underbrace{\underline{\sf{additional\;information:-}}}\end{gathered}$

ALGEBRAIC IDENTITIES -

Identity 1: (a + b)² = a² + 2ab + b²

Proof:

(a + b)² = (a + b) (a + b)

(a + b)² = a(a + b) + b(a + b)

(a + b)² = a² + ab + ba + b²

(a + b)² = a² + 2ab + b²

→ Hence, (a + b)² = a²+ 2ab + b²

Identity 2: (a -b)² = a² - 2ab + b²

Proof:

(a - b)² = (a - b) (a - b)

(a - b)² = a( a - b ) -b( a - b )

(a - b)² = a² - ab - ba + b²

(a - b)² = a² - 2ab + b²

→ Hence, (a – b)² = a² - 2ab + b2

Identity 3: a² - b² = (a + b) (a-b)

Identity 4: (x +a) (x+ b) = x² + (a + b)x + ab

Proof:

(x + a) (x + b) = x(x + b) + a(x + b)

(x + a) (x + b) = x² + bx + ax + ab

(x + a) (x + b) = x² + ax + bx + ab

(x + a) (x + b) =x² + (a + b)x + ab

→ Hence, (x + a) (x + b) = x² + (a + b)x + ab

Answer 2

a ) (3x + 5y) (3x + 5y)

→ 9x² + 30xy + 25y²

b ) (3x² + 2) (3x² + 2)

→ 9x⁴ + 12x² + 4

$\sf c) \: ( \dfrac{2}{3} x + \dfrac{3}{4} y)( \dfrac{2}{3} x + \dfrac{3}{4} y)$

→ $\sf c) \: \dfrac{4}{9} {x}^{2} + xy + \dfrac{9}{16} {y}^{2}$

$\sf d) \: ( \dfrac{3}{4} {x}^{2} + 5) ( \dfrac{3}{4} {x}^{2} + 5)$

→$\sf d) \: \dfrac{9}{16} {x}^{4} + \dfrac{15}{2} {x}^{2} + 25$