Given that for every value of x, x^2 - 10x + 39 = (x - a)^2 + b, then what is the value of b?

Question

Not a simplified version but a numerical answer.

Answer 1

This is completing the square.

 

From the standard equation,   x² - 10x + 39 , we convert this to vertex form.

 

(x² - 10x + 25) + 39 - 25 =

 

(x - 5)(x - 5) + 39 - 25 =

 

(x - 5)² + 14

 

 

b = 14

Answer 2

x² + 10x + 39 = (x - a)² + b

x² + 10x + 39 = x² - 2ax + a²+ b

10x = -2ax

-5 = a

a² + b = (-5)² + b

a² + b = 39

 

Can you solve for b?

 

Answer 3

Add and subtract (half the coefficient of x)^2

 

X^2 -10X +25 -25 +39

 

(X-5)^2 -25 +39

 

(X-5)^2 +14

 

Hence b = 14