To make it a perfect square we must add 81 to the expression.
Perfect square
A perfect square is a number that is the result of the product of another number by itself.
For example, 9 is a perfect square as it is the product of 3 and 3.
9 = 3 x 3
Given to us,
[tex]x^2+18x+.....[/tex]
at the blank space, there must be some constant so that the equation can be a perfect square.
We know that a perfect square is given as,
[tex](a+b)^2=a^2+2ab+b^2[/tex]
Comparing both the equations, we find out that
a² = x²,
2ab = 18x,
a²=x²,
a=x
substituting a's value in 2ab = 18x,
[tex]2\times a\times b= 18\times x\\2\times x \times b = 18\times x\\\\b = \dfrac{18\times x}{2\times x}\\\\b= 9[/tex]
therefore,
[tex]b=9\\b^2=9^2\\b^2 = 81[/tex]
Hence, to make it a perfect square we must add 81 to the expression.
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