Question

solve by completing the square.
( f^2 - 18f = 5 )
write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.
( f = square ) or ( f = square )
save answer

Explanation:

Step1: Complete the square for \(f^2 - 18f\)

To complete the square for the quadratic expression \(f^2 - 18f\), we take half of the coefficient of \(f\), which is \(\frac{-18}{2}=-9\), and then square it: \((-9)^2 = 81\). We add and subtract this value on the left - hand side of the equation \(f^2-18f = 5\). But since we are working with an equation, we add 81 to both sides to maintain equality.
The equation becomes \(f^2-18f + 81=5 + 81\).

Step2: Rewrite as a perfect square

The left - hand side \(f^2-18f + 81\) can be written as a perfect square \((f - 9)^2\) (using the formula \((a - b)^2=a^2-2ab + b^2\), where \(a = f\) and \(b = 9\)). The right - hand side is \(5+81 = 86\). So we have the equation \((f - 9)^2=86\).

Step3: Solve for \(f\)

Take the square root of both sides: \(f - 9=\pm\sqrt{86}\). Then, solve for \(f\) by adding 9 to both sides: \(f=9\pm\sqrt{86}\).
Calculate the numerical values: \(\sqrt{86}\approx9.27\). So \(f = 9+\sqrt{86}\approx9 + 9.27=18.27\) and \(f=9-\sqrt{86}\approx9 - 9.27=- 0.27\).

Answer:

\(f = 9+\sqrt{86}\) (or approximately \(18.27\)) or \(f = 9-\sqrt{86}\) (or approximately \(-0.27\))