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Solve for $f$ . $\newline$ $f^2 - 18f + 81 = 0$ $\newline$ Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas. $\newline$ $f =$ ____

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Solve polynomial equations

Full solution

Q. Solve for

f

.

\newline

f^2 - 18f + 81 = 0

\newline

Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.

\newline

f =

____

Factorize quadratic: First, let's see if the quadratic can be factored easily. The equation looks like a perfect square because $81$ is $9^2$ and $18$ is $2\times9$ .
Expand and simplify: So, we try $(f - 9)(f - 9) = f^2 - 9f - 9f + 81$ , which simplifies to $f^2 - 18f + 81$ .
Write in squared form: This matches our original equation, so we can write $f^2 - 18f + 81 = (f - 9)^2$ .
Set equation to zero: Now, set $(f - 9)^2 = 0$ and solve for $f$ .
Solve for $f$ : Taking the square root of both sides gives us $f - 9 = 0$ .
Final solution: Finally, add $9$ to both sides to get $f = 9$ .

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