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Sagot :
Solution
Given the expression below
Let the number be k
[tex]x^2-16x+k[/tex]To make it a perfect square, we apply the perfect square formula
[tex](x+a)^2[/tex]Equating both equations
[tex]\begin{gathered} x^2-16x+k=(x+a)^2 \\ x^2-16x+k=(x+a)(x+a) \\ x^2-16x+k=x(x+a)+a(x+a) \\ x^2-16x+k=x^2+ax+ax+a^2 \\ x^2-16x+k=x^2+2ax+a^2 \end{gathered}[/tex]Equating the terms
[tex]\begin{gathered} 2ax=-16x \\ \text{Divide both sides by 2x} \\ \frac{2ax}{2x}=\frac{-16x}{2x} \\ a=-8 \end{gathered}[/tex]Where the term added k is
[tex]\begin{gathered} k=a^2 \\ a=-8 \\ k=(-8)^2 \\ k=64 \end{gathered}[/tex]Substituting for k into the expression
[tex]\begin{gathered} x^2-16x+k \\ k=64 \\ x^2-16x+64 \end{gathered}[/tex]Hence, the trinomial is
[tex]x^2-16x+64[/tex]The number to be added is 64
Writing the trinomial as the square of a binomial becomes
[tex]\begin{gathered} x^2-16x+64 \\ =x^2-8x-8x+64 \\ =x(x-8)-8(x-8)_{} \\ =(x-8)(x-8) \\ =(x-8)^2 \end{gathered}[/tex]Hence, the square of the binomial is
[tex](x-8)^2[/tex]Thus, the number to be added is 64 and the square of the binomial is (x - 8)²
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