Question

If (x-1)(x+5)=(x-h)^2+K then what is the value of h

Answer 1

Solve for h:   (I'm using the completing the square)
(x - 1) (x + 5) = K + (x - h)^2
(x - 1) (x + 5) = K + (x - h)^2 is equivalent to K + (x - h)^2 = (x - 1) (x + 5):
K + (x - h)^2 = (x - 1) (x + 5)
Subtract K from both sides:
(x - h)^2 = (x - 1) (x + 5) - K
Take the square root of both sides:
x - h = sqrt((x - 1) (x + 5) - K) or x - h = -sqrt((x - 1) (x + 5) - K)
Subtract x from both sides:
-h = sqrt((x - 1) (x + 5) - K) - x or x - h = -sqrt((x - 1) (x + 5) - K)
Multiply both sides by -1:
h = x - sqrt((x - 1) (x + 5) - K) or x - h = -sqrt((x - 1) (x + 5) - K)
Subtract x from both sides:
h = x - sqrt((x - 1) (x + 5) - K) or -h = -x - sqrt((x - 1) (x + 5) - K)
Multiply both sides by -1:
Answer:  h = x - sqrt((x - 1) (x + 5) - K) or h = x + sqrt((x - 1) (x + 5) - K)


Solve for h: using the quadratic formula)
(x - 1) (x + 5) = K + (x - h)^2
(x - 1) (x + 5) = K + (x - h)^2 is equivalent to K + (x - h)^2 = (x - 1) (x + 5):
K + (x - h)^2 = (x - 1) (x + 5)
Expand out terms of the left hand side:
h^2 + K - 2 h x + x^2 = (x - 1) (x + 5)
Subtract (x - 1) (x + 5) from both sides:
h^2 + K - 2 h x + x^2 - (x - 1) (x + 5) = 0
h = (2 x ± sqrt(4 x^2 - 4 (K + x^2 - (x - 1) (x + 5))))/2:
Answer:  h = x + sqrt(-5 - K + 4 x + x^2) or h = x - sqrt(-5 - K + 4 x + x^2)