Completing the square
Initial explanation
We want to solve the following equation
x² + 16x + 5 = 0
↓
x² + 16x = 5
We want to rewrite the equation so its left side is
(x + a)²
We know that
(x + a)² = x² + 2ax + a²
where a is a number, then, we want that the left side look something like
x² + 2ax + a²
Then we first want to find which must be the number a in this case.
Finding a
Since 2 · 8 = 16, then the left side is
x² + 16x = x² + 2 · 8x
then, in this case a would be 8
a = 8
Completing the square with a²
We want that the left side of our equation looks like:
x² + 2ax + a²
Since it has the first two terms
x² + 16x = 5
we have to add a² = 8² = 64
We add it both sides of the equation:
x² + 16x + 8² = 5 + 64
x² + 2 · 8x + 8² = 69
Now, we can write it as (x + a)²
x² + 2 · 8x + 8² = 69
↓ since (x + a)² = x² + 2ax + a²
(x + 8)² = 69
Solving the equation
In order to solve the equation, we want to "leave x alone" on the left side:
(x + 8)² = 69
↓ squaring root both sides
(x + 8) = ±√69
x + 8 = ±√69
↓ taking 8 to the right side
x = - 8 ±√69
Solutions
We have two solutions:
x₁ = - 8 -√69
x₂ = - 8 + √69
Since