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  • Factorize : (i) x^{2}+9x+18 (ii) 6x^{2}+7x-3 (iii) 2x^{2}-7x-15 (iv)84-2r-2r^{2}

Factorize :

(i) x^{2}+9x+18

(ii) 6x^{2}+7x-3

(iii) 2x^{2}-7x-15

(iv)84-2r-2r^{2}

 

Answers (1)

(i)(x+6)(x+3)  

Solution

Given, x^{2}+9x+18

To factorize ax^{2} + bx + c, we have to distribute “bx” into “px” and “qx” such that

p + q = b and p.q = a.c

The given equation can be written as: 

x^{2}+9x+18\\ \Rightarrow x^{2}+3x+6x+18 \; \; \; \; \; \; \; \left \{ 6+3=9, (6)(3)=18 \right \}\\ \Rightarrow x(x+3)+6(x+3)\\ \Rightarrow (x+3)(x+6).

(ii) (2x+3)(3x-1)  

Solution

Given, 6x^{2}+7x-3

To factorize ax^{2} + bx + c, we have to distribute “bx” into “px” and “qx” such that

 p + q = b and p.q = a.c

 The given equation can be written as: 

     6x^{2}+7x-3\\ \Rightarrow 6x^{2}+9x-2x-3\; \; \; \; \; \; \; \left \{ 9-2=7, (9)(-2)=(6)(-3) \right \}\\ \Rightarrow (6x^{2}+9x)-(2x+3)\\ \Rightarrow 3x(2x+3)-1(2x+3)\\ \Rightarrow (2x+3)(3x-1).

(iii) (x-5)(2x-3)  

Solution

Given, 2x^{2}-7x-15

To factorize ax^{2} + bx + c, we have to distribute “bx” into “px” and “qx” such that

p + q = b and p.q = a.c

The given equation can be written as: 

2x^{2}-7x-15\\ \Rightarrow 2x^{2}-10x+3x-15\; \; \; \; \; \; \; \left \{ -10+3=-7, (-10)(3)=(2)(-15) \right \}\\ \Rightarrow (2x^{2}-10x)+(3x-15) \\ \Rightarrow 2x(x-5)+3(x-5)\\ \Rightarrow (x-5)(2x+3).

(iv) (6-r)(14-2r)  

Solution

Given, 84-2r-2r^{2}

To factorize ax^{2} + bx + c, we have to distribute “bx” into “px” and “qx” such that

p + q = b and p.q = a.c

The given equation can be written as: 

84-2r-2r^{2}\\ \Rightarrow 84-14r+12r-2r^{2}\; \; \; \; \; \; \; \left \{ -14+12=-2, (-14)(12)=(84)(-2) \right \}\\ \Rightarrow (84-14r)+(12r-2r^{2}) \\ \Rightarrow 14(6-r)+2r(6-r)\\ \Rightarrow (6-r)(14+2r).

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