Practice the questions given in the worksheet on completing square.
1. Write the following as a perfect square.
(i) 4X\(^{2}\) + 4X + 1
(ii) 9a\(^{2}\) – 12ab + 4b\(^{2}\)
(iii) 1 + \(\frac{6}{a}\) + \(\frac{9}{a^{2}}\)
2. Indicate the perfect squares among the following. Express each of the perfect squares as the square of a binomial. What numbers should be added to those which are not perfect squares so that the expressions may become perfect squares?
(i) 36x\(^{2}\) – 60xy + 25y\(^{2}\)
(ii) x\(^{2}\) + 4x + 1
(iii) 4a\(^{2}\) + 4a
(iv) 9a\(^{2}\) – 6a + 1
(v) 16 – 24a + 9a\(^{2}\)
(vi) 25x\(^{2}\) + 10x – 1
3. Find the missing term in each of the following so that the expression becomes a perfect square.
(i) 25x\(^{2}\) + (..........) + 49
(ii) 64a\(^{2}\) - (..........) + b\(^{2}\)
(iii) 9 + (..........) + x\(^{2}\)
(iv) 16a\(^{2}\) + 8a + (..........)
(v) (..........) – 18x + 9x\(^{2}\)
(vi) x\(^{2}\) – 2 + (..........)
4. Each of the following is a perfect square. Find the numerical value of k.
(i) 121a\(^{2}\) + ka + 1
(ii) 3ka\(^{2}\) + 24a + 4
[Hint: 3ka\(^{2}\) + 2 ∙ 6a ∙ 2 + 2\(^{2}\). So, 3ka\(^{2}\) = (6a)\(^{2}\). Therefore, 3k = 6\(^{2}\)]
(iii) 4x\(^{4}\) + 12x\(^{2}\) + k
5. What should be added to make each of the following a perfect square?
(i) 25x\(^{2}\) + 81
(ii) 81x\(^{2}\) – 18x
(iii) a\(^{4}\)+ \(\frac{1}{a^{4}}\)
Answers for the worksheet on completing square are given below.
Answer:
1. (i) (2x + 1)\(^{2}\)
(ii) (3a – 2b)\(^{2}\)
(iii) (1 + \(\frac{3}{a}\))\(^{2}\)
2. (i) Perfect square, (6x – 5y)\(^{2}\)
(ii) Not a perfect square, 3
(iii) Not a perfect square, 1
(iv) Perfect square, (3a - 1)\(^{2}\)
(v) Perfect square, (4 – 3a)\(^{2}\)
(vi) Not a perfect square, 2
3. (i) 70x
(ii) 16ab
(iii) 6x
(iv) 1
(v) 9
(vi) \(\frac{1}{x^{2}}\)
4. (i) 22
(ii) 12
(iii) 9
5. (i) 90x
(ii) 1
(iii) 2 or -2
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