Solve by completing the square.
k^2 − 16k + 3 = 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 need to take the middle term, divide it by 2, square the result, and then add and subtract that value within the equation.
First, we rewrite the equation by adding and subtracting the square of half the coefficient of the middle term:
k^2 - 16k + 64 - 64 + 3 = 0
(k - 8)^2 - 61 = 0
Now we can solve for k by isolating (k - 8)^2:
(k - 8)^2 = 61
Now we take the square root of both sides:
k - 8 = ±√61
k = 8 ±√61
So, the solutions to the equation k^2 − 16k + 3 = 0 are:
k = 8 + √61 or k = 8 - √61