How do you solve using the completing the square method #x^2 - 12x + 36 = 25 #?

Question

Nathan Axel · Accepted Answer

The solutions are:
#color(blue)(x=11#

#color(blue)(x=1# #x^2 -12x +36=25# Using the Identity #color(blue)((a-b)^2 = a^2 - 2ab + b^2#, we get #(x-6)^2 = 25# #(x-6) =+-sqrt 25# #(x-6) =+5# #x= 5 +6# #color(blue)(x=11# #(x-6) =-5# #x=-5+6# #color(blue)(x=1#

Eva Abramson · Answer

undefined To solve using the completing the square method, follow these steps:

1. Move the constant term to the other side of the equation:
   x^2 - 12x + 36 - 25 = 0

2. Combine like terms:
   x^2 - 12x + 11 = 0

3. To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation:
   x^2 - 12x + (12/2)^2 = 11 + (12/2)^2
   x^2 - 12x + 36 = 11 + 36

4. Simplify both sides:
   x^2 - 12x + 36 = 47

5. Factor the left side:
   (x - 6)^2 = 47

6. Take the square root of both sides:
   x - 6 = ±√47

7. Solve for x:
   x = 6 ± √47

Genesis Allen · Answer

undefined To solve the quadratic equation $ x^2 - 12x + 36 = 25 $ using the completing the square method:

1. Move the constant term to the other side of the equation:
   $$ x^2 - 12x + 36 - 25 = 0 $$

2. Combine like terms:
   $$ x^2 - 12x + 11 = 0 $$

3. Rewrite the equation in the form $ (x - h)^2 = k $ by completing the square.
   $$ x^2 - 12x + 36 = 25 $$
   $$ x^2 - 12x + (12/2)^2 = 25 + (12/2)^2 $$
   $$ x^2 - 12x + 36 = 25 + 36 $$

4. Simplify both sides:
   $$ (x - 6)^2 = 61 $$

5. Take the square root of both sides:
   $$ x - 6 = \pm \sqrt{61} $$

6. Solve for $ x $:
   $$ x = 6 \pm \sqrt{61} $$

So, the solutions to the equation $ x^2 - 12x + 36 = 25 $ using the completing the square method are $ x = 6 + \sqrt{61} $ and $ x = 6 - \sqrt{61} $.