Solve by completing the square.
f^2 + 20f − 1 = 0
Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.
To solve by completing the square, we first move the constant term to the right side of the equation:
f^2 + 20f = 1
Next, we take half of the coefficient of f (which is 10) and square it to get the term we need to add and subtract to complete the square. So, we add (10)^2 = 100 to both sides:
f^2 + 20f + 100 = 101
Now we can rewrite the left side as a squared binomial:
(f + 10)^2 = 101
Taking the square root of both sides:
f + 10 = ±√101
f = -10 ± √101
So, the solutions are:
f = -10 + √101 or f = -10 - √101
These cannot be simplified any further, so the answers are:
f = -10 + √101 or f = -10 - √101