Respuesta :
Hello there. To solve this question, we'll have to remember some properties about completing the square.
Given the equation:
[tex]x^2+18x+90=0[/tex]To complete the square, we have to add a term on both sides of the equation in order to find a "perfect trinomial", that is the expanded version of a binomial (a + b)².
Knowing that (a + b)² = a² + 2ab + b², we start looking at the middle term and dividing its coefficient by 2: the middle term is 18x, the coefficient is 18.
Dividing it by 2, we get:
[tex]\frac{18}{2}=9[/tex]Now we add this value, squared, on both sides of the equation:
[tex]\begin{gathered} 9^2=81 \\ x^2+18x+90+81=0+81 \end{gathered}[/tex]In this case, we get that:
[tex]\begin{gathered} x^2+18x+81+90=81 \\ (x+9)^2+90=81 \end{gathered}[/tex]Subtract 81 on both sides of the equation
[tex]\begin{gathered} (x+9)^2+90-81=81-81 \\ (x+9)^2+9=0 \end{gathered}[/tex]This is the answer we're looking for.
