| What should be added to the \[x\,\,(x+a)(x+2a)\] \[(x+3a),\] so that the sum be a perfect square? |
A) \[{{a}^{2}}\]
B) \[{{a}^{4}}\]
C) \[{{a}^{3}}\]
D) \[{{a}^{6}}\]
Correct Answer: B
Solution :
| \[x\,\,(x+a)(x+2a)(x+3a)\] |
| \[=({{x}^{2}}+ax)({{x}^{2}}+5ax+6{{a}^{2}})\] |
| \[={{x}^{4}}+a{{x}^{3}}+5a{{x}^{3}}+5{{a}^{2}}{{x}^{2}}+6{{a}^{2}}{{x}^{2}}+6{{a}^{3}}x\] |
| \[={{x}^{4}}+ax\,\,({{x}^{2}}+5{{x}^{2}}+5ax+6ax+6{{a}^{2}})\] |
| \[={{x}^{4}}+ax\,\,(6{{x}^{2}}+11ax+6{{a}^{2}})\] (i) |
| For terms to be perfect square, |
| \[{{(x+y)}^{2}}{{(x+y)}^{2}}\] |
| \[=({{x}^{2}}+2xy+{{y}^{2}})({{x}^{2}}+{{y}^{2}}+2xy)\] |
| \[={{x}^{4}}+2{{x}^{3}}y+{{x}^{2}}{{y}^{2}}+{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}+{{y}^{4}}\] |
| \[+\,\,2{{x}^{3}}y+4{{x}^{2}}{{y}^{2}}+2x{{y}^{3}}\] |
| \[={{x}^{4}}+xy\,\,(4{{x}^{2}}+6xy+4{{y}^{2}})+{{y}^{4}}\] (ii) |
| On comparing Eqs. (i) and (ii), \[y=a\] |
| So, \[{{a}^{4}}\] must be added to make it a perfect square. |
You need to login to perform this action.
You will be redirected in
3 sec
