Question

Simplify the following using the identity
$\tiny(x + a)(x + b) = {x}^{2} + x(a + b) + ab: \\\ \textless \ br /\ \textgreater \ (i)(x+3)(x+5) \\\ \textless \ br /\ \textgreater \ (ii)(y+8)(y+6) \\\ \textless \ br /\ \textgreater \ (iii)43×36$
No copied answers.​

Answer 1

Solution:-

(i) (x+3)(x+5)

Let us represent the expression giometrically, as shown in the fig. 3.11.

In the rectangle with length (x+5) and breadth (x+3),

We get,

$\small\tiny\bold \color{navy} \: area \: of \: bigger \: rectangle \: = area \: of \: square + area \: of \: three \: rectangles$

Therefore,

$\tiny(x + 3)( x+ 5) = {x}^{2} + 5x + 3x + 15$

$\tiny \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {x}^{2} +( 5 + 3)x + 15 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tiny {x}^{2} + 8x + 15.$

(ii) (y+6)(y+8)

$\color {purple}\tiny \sf \underline {let \: us \: represent \: the \: expression \: geometrically} \\ \color{purple} \tiny \sf \underline {as \: shown \: in \: fig.3.12. \: in \: the \: rectangle \: with \: length(y +6 )} \\ \color{purple} \sf \tiny \underline{and \: breadth \: (y + 8)units}$

We get,

$\color {navy} \tiny \sf \underline {area \: of \: bigger \: rectangle \: = area \: of \: square + area \: of \: three }\: \\ \tiny\sf \color{navy} \underline {rectangles}$

Therefore,

$\tiny(y + 6)(y + 8) = {y}^{2} + 6y + 8y + 48$

$\tiny(y + 6)( y+ 8) = {y}^{2} + (6 + 8)y + 48$

$\tiny \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = {y}^{2} + 14y + 48$

(iii) 43×36=(40+3)×(40-4)

We know the identity,

$\tiny(x + a)(x + b) = {x}^{2} + x(a + b) + ab$

$\color{orange} \tiny \sf \: taking \: x = 40 \: and\: \: a = 3 \: and \: b = - 4 \:and \: we \: get$

$\tiny(40 + 30)(40 - 4) = {40}^{2} + 40 (3 - 4) + 3( - 4)$

$\sf\tiny= 1600 + 40( - 1) - 12 \\ \sf \tiny = 1600 - 40 - 12 \\ \sf \tiny = 1600 - 52$

Therefore,

$\color{navy}43 \times 36 = 1548$

$\huge \sf\color{red} \star{hope \: it \: helps}$