Completing the square
Initial explanation
We want to convert
x² + 8x + 9 = 0
into an equation of the form:
(x + a)² = b
We have that
(x + a)² = x² + 2· ax + a²
Step 1
We are going to take the last term to the right side of the equation:
x² + 8x + 9 = 0
↓
x² + 8x = -9
Step 2
We are going to transform the second term into a multiplication so we can find a:
x² + 8x = -9
↓ since 8x = 2 · 4x
x² + 2 · 4x = -9
Since
(x + a)² = x² + 2· ax + a²
observing the second term:
2 · 4x = 2· ax
Then, in this case
4 = a
Step 3
Completing the square:
We want that the left side of our equation
x² + 2 · 4x = -9
have the same form as:
x² + 2· ax + a²
Since a = 4 and a² = 4² = 16
for it to happen, we need to add 16 to the equation:
x² + 2 · 4x = -9
↓
x² + 2 · 4x + 16 = -9 + 16 = 7
x² + 2 · 4x + 4² = 7
Now, we can factor the left side into a square:
x² + 2 · 4x + 4² = (x + 4)²
Then,
x² + 2 · 4x + 4² = 7
↓
(x + 4)² = 7
Answer - A. (x + 4)² = 7