### completing the square

Question Sample Titled 'completing the square'

 (a) Let ${f{{\left({x}\right)}}}={98}{x}-{x}^{{2}}$ . Using the method of completing the square, find the coordinates of the vertex of the graph of ${y}={f{{\left({x}\right)}}}$ . (2 marks) (b) The figure below shows a pattern made by some metal wires, where ${A}{B}{C}{D}$ is a rectangle. All metal wires inside ${A}{B}{C}{D}$ are parallel to ${B}{C}$ . It is given that the total length of the metal wires is ${686}$ $\text{cm}$ .

${A}$${B}$${C}$${D}$

 (i) Let ${x}$ $\text{cm}$ be the length of ${B}{C}$; and ${A}$ $\text{cm}^{{2}}$ be the area of ${A}{B}{C}{D}$. Express ${A}$ in terms of ${x}$. (ii) Muhammad claims that the area of ${A}{B}{C}{D}$ can be greater than ${9000}$ $\text{cm}^{{2}}$ . Do you agree? Explain your answer.  (4 marks)

 (a) ${f{{\left({x}\right)}}}$ $={98}{x}-{x}^{{2}}$  $=-{\left({x}^{{2}}-{98}{x}\right)}$  $=-{\left({x}^{{2}}-{98}{x}+{49}^{{2}}-{49}^{{2}}\right)}$ 1M  $=-{\left({x}-{49}\right)}^{{2}}+{2401}$ Required coordinates $={\left({49},{2401}\right)}$ 1A  (bi) ${A}$ $={x}{\left(\dfrac{{{686}-{7}{x}}}{{2}}\right)}$ 1M  $={343}{x}-\dfrac{{7}}{{2}}{x}^{{2}}$ 1A  (bii) Note that ${A}$ $=\dfrac{{7}}{{2}}{f{{\left({x}\right)}}}$ , where ${f{{\left({x}\right)}}}={98}{x}-{x}^{{2}}$ and ${0}<{x}<{98}$ . By (a) , the greatest value of ${A}$ is ${8403.5}$ $<{9000}$. 1M Thus, he is disagreed. 1A

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