Question

What is the missing constant term and a perfect square starts with x^2+10x

Answer 1

The perfect square is a number that is the product of two identical numbers. So 9 is a perfect square because it's the product of 3 * 3. For variables, perfect squares can be expressed as $(x+a)^{2}$ or $(x-a)^{2}$, where a is some number and x is the variable.
If we expand these terms, $(x+a)^{2}=x^{2}+2ax+a^{2}$, and $(x-a)^{2}=x^{2}-2ax+a^{2}$. Since the given term is $x^{2}+10x$, we can see that it's part of the perfect square $x^{2}+2ax+a^{2}$. We just need to find $a^{2}$.
So, $x^{2}+2ax+a^{2}=x^{2}+10x+a^{2}$. Notice that $10=2a$, so $a=5$. So, the missing constant term is $5^{2}=25$. The perfect square is $(x+5)^{2}$.