Question

Solve (p-4)^2=9p using completing the square

Answer 1

First, expand the left side of the equation:

(p-4)(p-4) = 9p
p^2 - 8p + 16 = 9p

Next, move all terms to one side of the equation:

p^2 - 8p - 9p + 16 = 0
p^2 - 17p + 16 = 0

To complete the square, halve the coefficient of the linear term, square it, and add and subtract the result inside the parentheses:

(p - (17/2))^2 - (17/2)^2 + 16 = 0
(p - (17/2))^2 - 289/4 + 16 = 0
(p - (17/2))^2 - 273/4 = 0

Now, solve for p:

(p - (17/2))^2 = 273/4
p - (17/2) = ±√(273/4)
p = (17/2) ± √(273)/2
p = (17 ± √273)/2

Therefore, the solutions to the equation are p = (17 + √273)/2 and p = (17 - √273)/2.