$852$ . Поставьте вместо знака $$ такой одночлен, чтобы трёхчлен но было представить в виде квадрата двучлена: $\newline$ а) $+56 a+49$ ; $\newline$ б) $36-12 x+$ ; $\newline$ в) $25 a^{2}++\frac{1}{4} b^{2}$ ; $\newline$ г) $0,01 b^{2}+*+100 c^{2}$ .

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852

. Поставьте вместо знака

*

такой одночлен, чтобы трёхчлен но было представить в виде квадрата двучлена:

\newline

а)

*+56 a+49

;

\newline

б)

36-12 x+*

;

\newline

в)

25 a^{2}+*+\frac{1}{4} b^{2}

;

\newline

г)

0,01 b^{2}+*+100 c^{2}

Recognize Perfect Square Trinomial: To represent a trinomial as the square of a binomial, the trinomial must be a perfect square trinomial. The general form of a perfect square trinomial is $(a + b)^2 = a^2 + 2ab + b^2$ . We need to find the missing term that will complete the square for each case.
Complete the Square: $49$ : а) For the trinomial $x^2 + 56a + 49$ , we recognize that $49$ is a perfect square, $7^2$ . The term $56a$ suggests that the binomial form could be $(x + 7)^2$ , where $x$ is the variable part of the first term. To complete the square, we need the first term to be $(7x)^2$ , so $x$ should be $a$ . Therefore, the missing term is $7a \times 7a = 49a^2$ .
Complete the Square: $36$ : б) For the trinomial $36 - 12x + **$ , we recognize that $36$ is a perfect square, $6^2$ . The term $-12x$ suggests that the binomial form could be $(6 - x)^2$ . To complete the square, we need the last term to be $(-x)^2$ , so the missing term is $x^2$ .
Complete the Square: $25a^2$ : в) For the trinomial $25a^2 + ** + (\frac{1}{4})b^2$ , we recognize that $25a^2$ is a perfect square, $(5a)^2$ , and $(\frac{1}{4})b^2$ is also a perfect square, $(\frac{1}{2}b)^2$ . The middle term should be $2\cdot(5a)\cdot(\frac{1}{2}b) = 5ab$ . Therefore, the missing term is $5ab$ .
Complete the Square: $0.01b^2$ : г) For the trinomial $0.01b^2 + ** + 100c^2$ , we recognize that $0.01b^2$ is a perfect square, $(0.1b)^2$ , and $100c^2$ is also a perfect square, $(10c)^2$ . The middle term should be $2\times(0.1b)\times(10c) = 2bc$ . Therefore, the missing term is $2bc$ .