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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 101. |
Factorize `(i) (x^2-(y^2)/(100))` `(ii)(100-9x^2)` `(iii)(49x^2-(1)/(4))` |
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Answer» We have ` (i) (x^2-(y^2)/(100))=x^2-((y)/(10))^2` `therefore (x^2-(y^2)/(100))=(x-(y)/(10))(x+(y)/(10))`. ` (ii) (100-9x^2)=(10)^2-(3x)^2` `=(10-3x)(10+3x)` `therefore (100-9x^2)=(10-3x)(10+3x)`. `(iii) (49x^2-(1)/(4))=(7x)^2-((1)/(2))^2` `=(7x-(1)/(2)) (7x+(1)/(2))` `therefore (49x^2-(1)/(4)) =(7x-(1)/(2))(7x+(1)/(2))`. |
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| 102. |
Find the roots of `x^2+18x+32` if they exist |
| Answer» Correct Answer - `(x+16)(x+2)` | |
| 103. |
Factroize `(i) (x^3-x)` `(ii) 18a^2x^2-32)` `(iii) (2a^5-32a)`. |
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Answer» We have `(i) (x^3-x)=(x^2 -1)` `=x(x-1)(x+1)` `therefore(x^3-x)=x(x-1)(x+1)`. `(ii) (18a^2x^2-32) =2(9a^2x^2-16)` ` =2xx[(3ax)^2-4^2]` ` =2(3ax-4)(3ax+4)` `therefore (18a^2x^2-32)=2(3ax-4)(3ax+4)`. `(iii) (2a^5-32a)=2a(a^4-16)` `=2axx [(a^2)^2-4^2]` ` =2a(a^2-4)(a^2+4)` ` =2a (a-2)(a+2)(a^23+4)` ` therefore (2a^5-32a)=2a(a-2)(a+2)(a^2+4)` |
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| 104. |
`(4x^2+4x-3)=?`A. `(2x-1)(2x-3)`B. `(2x+1)(2x-3)`C. `(2x+3)(2x-1)`D. None of these |
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Answer» Correct Answer - C `(4x^2+4x-3)=(4x^2+6x-2x-3) =2x(2x+3)-(2x+3)=(2x+3)(2x-1)`. |
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| 105. |
`2x^2+11x-21` |
| Answer» Correct Answer - `(x+7)(2x-3)` | |
| 106. |
`(305xx 308)=?`A. `94940`B. `93840`C. `93940`D. `94840` |
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Answer» Correct Answer - C `(305xx308)=305xx (300+8)`. `=(305xx 200)-(305xx 8) =(91500+2440)=93940`. |
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| 107. |
`(x^2-4x-21)=?`A. `(x-7)(x-3)`B. `(x+7)(x-3)`C. `(x-7)(x+3)`D. None of these |
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Answer» Correct Answer - C `(x^2-4x-21)=(x^2-7x+3x-21)`. `=x(x-7)+3(x-7)=(x-7)(x+3)`. |
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| 108. |
`(207xx193)=?`A. `39851`B. `39951`C. `93940`D. `38951` |
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Answer» Correct Answer - B `(207xx 193)=207xx(200-7)` ` =(207xx200)-(207xx7)=(41400-1449)=39951`. |
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| 109. |
`4a^2+b^2+4ab+8a+4b+4= ? `A. `(2a+b+2)^2`B. `(2a-b+2)^2`C. `(a+2b+2)^2`D. None of these |
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Answer» Correct Answer - A Given expression `=(2a)^2+b^2+2^2+2xx[(2a)xx b+(bxx2)+(2axx 2)]=(2a+b+2)^2`. |
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| 110. |
` 2 sqrt3 x^2 + x - 5sqrt3` |
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Answer» Correct Answer - `(x+sqrt(3))(2sqrt(3)x-5)` `2sqrt(3)x^2+x-5sqrt(3)=2sqrt(3)x^2+6x-5x-5sqrt(3)` |
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| 111. |
`x^2-7sqrt(6)x-48` |
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Answer» Correct Answer - `(x+5sqrt(6))(x+2sqrt(6))` `x^2+7sqrt(6)x+60=x^2+5sqrt(6)x+2sqrt(6)x+60`. |
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| 112. |
`x^2-2sqrt(3)x-24` |
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Answer» Correct Answer - `(x-4sqrt(3))(x+2sqrt(3))` `x^2-2sqrt(3)x-24=x^2-4sqrt(3)x+2sqrt(3)x-24`. |
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| 113. |
`x^2-3sqrt(5)x-20` |
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Answer» Correct Answer - `(x-4sqrt(5))(x+sqrt(5))` `x^2-3sqrt(5)x-20=x^2-4sqrt(5)x +sqrt(5)x-20`. |
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| 114. |
`x^2+3sqrt(3)x+6` |
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Answer» Correct Answer - `(x+2sqrt(3))(x+sqrt(3))` `x^2+3sqrt(3)x+6=x^2+2sqrt(3)x+sqrt(3)x+6`. |
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| 115. |
`x^2+sqrt(2)x-24` |
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Answer» Correct Answer - `(x+4sqrt(2))(x-3sqrt(2))` `x^2+sqrt(2)x-24=x62+4sqrt(2)x-3sqrt(2)x-24`. |
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| 116. |
`6sqrt3 x^2 - 47 x + 5 sqrt3` |
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Answer» Correct Answer - `(2x-5sqrt(3))(3sqrt(3)x-1)` `6sqrt(3)x^2 -47x+5sqrt(3)=6sqrt(3)x^2-45x-2x+5sqrt(3)=3sqrt(3)x(2x-5sqrt(3))-(2x-5sqrt(3))`. |
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| 117. |
`(3)/(5)x^2-(19)/(5)x+4` |
| Answer» Correct Answer - `((x)/(5)-1)(3x-4)` | |
| 118. |
`sqrt(5)x^2+2x-3sqrt(5)` |
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Answer» Correct Answer - `(x+sqrt(5))(sqrt(5)x-3)` `sqrt(5)x^2+2x-3sqrt(5)=sqrt(5)x^2+5x-3x-3sqrt(5)`. |
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| 119. |
Factorize:`x^2-2 sqrt(2) x-30` |
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Answer» Correct Answer - `(x-5sqrt(2))(x+3sqrt(2))` `x^2 -2sqrt(2)x-30=x^2+-5sqrt(2)x+3sqrt(2)x-30`. |
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| 120. |
Factorize:`5 sqrt(5) x^2+20 x+3 sqrt(5)` |
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Answer» Correct Answer - `(sqrt(5)x+3)(sqrt(2)x+1)` `5sqrt(5)x^2+20x+3sqrt(5)=5sqrt(5)x^2+15x+5x+3sqrt(5)=5x(sqrt(5)x+3)+sqrt(5)(sqrt(5)x+3)`. |
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| 121. |
Solve each of the following quadratic equations: `2x^(2)-x+(1)/(8)=0` |
| Answer» Correct Answer - `((x)/(2)-(1)/(8))(4x-1)` | |
| 122. |
If `(a)/(b)+(b)/(a)=1` then `(a^3-b^3)=?`A. `-3`B. `-2`C. `-1`D. `0` |
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Answer» Correct Answer - D `(a)/(b)+(b)/(a)=-1rArr a^2+b^2+ab=0` `rArr (a-b)(a^2-b^2+ab)=0rArr a^3-b^3=0`. |
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| 123. |
`sqrt(2)x^2+3x+sqrt(2)` |
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Answer» Correct Answer - `(x+sqrt(2))(sqrt(2)x+1)` `sqrt(2)x^2+3x+sqrt(2)=sqrt(2)x^2+2x+x+sqrt(2)`. |
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| 124. |
`2(x+y)^2-9(x+y)-5` |
| Answer» Correct Answer - `(x+y-5)(2x+2y+1)` | |
| 125. |
Factorize `(x^2+1/x^2-2-3x+3/x) ` |
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Answer» Correct Answer - `(x-(1)/(x))(x-(1)/(x)-3)` Given expression ` =(x-(1)/(x))^2-3(x-(1)/(x))`. |
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| 126. |
Expand `(2a+3b+4c)^2`. |
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Answer» Putting `2a-x,3b=y and 4c=z`, we get `(2a+3b+4c)^2` `=(x+y+z)^2` `=x^2+y^2+z^2+2xy+2yz+2zx` ` =(2a)^2+(3b)^2+(4c)^2+{2xx (2a) xx (3b)}+{2xx(3b)xx(4c)}+{2xx (4c)xx (2a)}` `=4a62+9b^2+16c^2+12ab+24bc+2zx`. |
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| 127. |
`2x^2 + 3 sqrt3x + 3` |
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Answer» Correct Answer - `(x+sqrt(3))(2x+sqrt(3))` `2x^2+3sqrt(3)x+3=2x^2+2sqrt(3)x+sqrt(3)x+3`. |
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| 128. |
Factorize (i) ` 5a^2-20ab` (ii) ` 36a^3b-60a^2bc` (iii) `5a(b+c)-7b(b+c)` (iv)` 6(2a+3b)^2-8(2a+3b)` |
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Answer» (i) ` 5a^2--20ab =5a(a-4b)` (ii) ` 36a^3b-60a^2bc=12a^2b(3a-5c)`. (iii) ` 5a (b+c) -7b(b+c)=(b+c) (5a-7b)`. (iv) ` 6(2a+3b)^2-8(2a+3b)=2(2a+3b)[3(2a+3b)-4]=2(2a+3b)(6a+9b-4)`. |
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| 129. |
`a(a-2b-c)+2bc` |
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Answer» Correct Answer - `(a-c)(a-2b)` Given expression ` =(a^2-ac)-(2ab-2bc)`. |
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| 130. |
`9(2a-b)^2-4(2a-b)-13` |
| Answer» Correct Answer - `(18a-9b-13)(2a-b+1)` | |
| 131. |
`a^3(b-c)^3+b^3(c-a)^3+c^3(a-b)^3` |
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Answer» Correct Answer - `3abc(b-c)(c-a)(a-b)` `a(b-c)+b(c-a)+c(a-b)=0` `rArr a^3(b-c)^3+b^3(c-a)^3+c^3(a-b)^3=3abc(b-c)(a-b)`. |
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| 132. |
`15x^2-x-28` |
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Answer» Correct Answer - `(5x-7)(3x+4)` `15x^2-x-28=15x^2-21x+20x-28`. |
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| 133. |
`2a(x+y)-3b(x+y)` |
| Answer» Correct Answer - `(x+y)(2a-3b)` | |
| 134. |
`x^(3)-x^(2)+ax+x-a-1` |
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Answer» Correct Answer - `(x-1)(x^2+a+1)` Given expression `=(x^3-x^2)+(ax-a)+(x-1)`. |
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| 135. |
`(a x+b y)^2+(b x-a y)^2` |
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Answer» Correct Answer - `(a^2+b^2)(x^2+y^2)` Given expression `=a^2x^2+b^2y^2+b^2 x^2+a^2y^2` `=(a^2x^2+b^2x^2)+(a^2y^2+b^2y^2)` . |
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| 136. |
`x^2+y^2-z^2-2xy` |
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Answer» Correct Answer - `(x-y+z)(x-y-z)` `x^2+y^2-z^2-2xy=(x^2+y^2-2xy)-z^2=(x-y)^2-z^2`. |
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| 137. |
Evaluate `(i) (-12)^3+7^3+5^3` `(ii) (28)^3+(-15)^3+(-13)^3` |
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Answer» Correct Answer - 108 (ii) Let ` a=28,b=-15 and c =-13`. Then, ` a+b+c=0 rArr a^3+b^3+c^3=3abc=3xx28xx (-15)xx(-13)=16380`. |
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| 138. |
`x^2-2x+(7)/(16)` |
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Answer» Correct Answer - `((x)/(4)-(7)/(16))(4x-1)` `x^2-2x+(7)/(16)=(16x^2-32x+7)/(16)=(16x^2-28x-4x+7)/(16)=((4x-7)(4x-1))/(16)=((4x-7)/(16))(4x-1)=((x)/(4)-(7)/(16))(4x-1)`. |
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| 139. |
`2x+4y-8xy-1` |
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Answer» Correct Answer - `(1-4y)(2x-1)` Given expression `=(2x-8xy)-1(1-4y)`. |
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| 140. |
`(a+b)^3-a-b` |
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Answer» Correct Answer - `(a+b)(a+b+1)(a+b-1)` `(a+b)^3-a-b=(a+b)^3-(a+b)=(a+b)[(a+b)^2 -1] =(a+b)[(a+b)62-1^2]` . |
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| 141. |
Evaluate `{(999)^2-1}` . |
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Answer» Correct Answer - `998000` `{(999)^2-1^2}=(999-1)(999+1)`. |
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| 142. |
evaluate using suitable identities1 .` (99)^3` 2. ` (103)^3` |
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Answer» Correct Answer - `(i) 1092727` `(ii) 970299` |
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| 143. |
Evaluate `(i) (99)^2` `(ii) (995)^2` `(iii) (107)^2` |
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Answer» Correct Answer - `(i) 9801` `(ii) 990025` `(iii) 11449` |
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| 144. |
`4a^2-4b^2+4a+1` |
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Answer» Correct Answer - `(2a+2b +1)(2a-2b+1)` `4a^2-4b^2+4a+1=(4a^2+4a+1)-4b^2=(2a+1)^2-(2b)^2`. |
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| 145. |
`2x(p^2+q^2)+4y(p^2+q^2)` |
| Answer» Correct Answer - `2(p^2+q^2)(x+2y)` | |
| 146. |
Evaluate `(i) (102)^3` `(ii) (999)^3` |
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Answer» We have `(i) (102)^3=(100+2)^3` `=(100)^3+2^3+3xx 100 xx 2 xx (100 +2)` `=1000000+8(600 xx 102)`. `=1000008 +61200 =1061208`. `(ii) (999)^3=(1000-1)^3` `=(1000)^3-1^3-(3xx 1000xx 1)(1000-1)` `=1000000000-1-(3000xx 999)` `=999999999-2997000=997002999`. |
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| 147. |
find the following products `(i) ((x)/(2)+2y)((x^(2))/(4)-xy+4y^(2))`(ii) `(x^(2)-1)(x^(4)+x^(2)+1)` |
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Answer» We have `(i) ((x)/(2)+2y)((x^2)/(4)-xy+4y^2)` `=(a+b)(a^2-ab)+b^2)," where " (x)/(2)=a and 2y=b` `=(a^3+b^3)` `=((x)/(2))^3+(2y)^3=((x^3)/(8)+8y^3)`. ` (ii) (x^2-1) (x^4+x^2+1)` ` =(a-b(a^2+ab+b^2), " where " x^2=a and 1=b` ` =(a^3-b^3)` `=(x^2)^3-(1)^3=(x^6-1)`. |
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| 148. |
Factroise `(i) 1+64x^3` `(ii) 27x^3+125y^3` `(iii) (x+1)^3+(x-1)^3` |
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Answer» We have `(i) 1+64x^3` `=(1)^3+(4x)^3` `=(1+4x)(1-4x+16x^2)` `(ii) 27x^3+125y^3` `=(3x)^3+(5y)^3` `=(3x+5y)(9x^2-15xy+25y^2)` (iii) Putting `(x+1)=a and (x-1)=b`, we get `(x+1)^3+(x-1)^3` `=(a^3+b^3)` ` =(a+b)(a^2-ab+b^2)` ` ={(x+1)+(x-1)}xx {(x+1)^2-(x+1)(x-1)+(x-1)^2}` ` =2xxx{(x^2+2x+1)-(x^2-1)+(x^2-2x+1)}` ` =2xx(x^2+2x+1-x^2+1+x^2-2x+1)=2x(x^2+3)`. |
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| 149. |
Factroise `(i) a^3+b^3+a+b` `(ii) a^3-b^3-a+b` |
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Answer» We have `(i) a^3+b^3+a+b` `=(a^3+b^3)+(a+b)` `=(a+b)(a^2-ab+b^2)+(a+b)` `=(a+b)xx{(a^2-ab+b^2)+1}` `=(a+b)(a^2-ab+b^2+1)`. `therefore (a^3+b^3+a+b)=(a+b)(a^2-ab+b^2+1)`. `(ii) a^3-b^3-a+b` `=(a^3-b^3)-(a-b)` `=(a-b)(a^2+ab+b^2)-(a-b)` `=(a-b)(a^2+ab+b^2 -1)`. `therefore (a^3-b^3-a+b)=(a-b)(a^2+ab+b^2-1)`. |
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| 150. |
Factorize `9x^2+4y^2+16z^2+12xy-16yz-24xz`. |
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Answer» We observe here that `-16yz and -24 xz` are negative terms in the given expression and z occruws in both the terms . So, we write ` 16z^2 "as " (-4z)^2`. Thus , we have `9x^2+4y^2-4xy-12yz+6xz` ` =(3x)62+(2y)^2+(-4z)^2+{(2xx(3x)xx (2y)}+{2xx(2y)xx(-4z)}+{2xx(-4z)xx(3x)}` `={3x+2y+(-4z)^2}=(3x+2y-4z)^2`. |
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