Solve the quadratic equation by completing the square

Question

Solve

x²+10x+16=0

Answer 1

x² + 10x + 16 = 0 Move the 16 to the other side by subtraction of 16 from both sides..

x² + 10x = -16 Coefficient of 1st term is already 1, so no additional operation required there. (Otherwise would have to force the first coefficient to 1 by dividing it away).

Now take 1/2 of the x coefficient and square it: 10/2 = 5 -> 5 x 5 = 5²

Add 5² to both sides: x² + 10x + 5² = -16 + 5²

Left side will become (x + 5)² (just use the 5 as the 2nd term inside the parentheses); right side is -16 + 25 = 9

Now, need to get rid of the squares by taking square root of both sides:

x + 5 = ±3 (note the ±) Subtract 5 from both sides: x = -5 + 3 = -2 or x = -5 - 3 = -8.

You can check by substituting both answers in the original quadratic.

Incidentally, the quadratic could have easily factored without completing the square, because factors of 16 are 8 and 2, hence (x + 2)(x + 8) = 0, etc.

Answer 2

We can add 9 to both sides to make the left side a square:

x² + 10x + 16 + 9 = 9

x² + 10x + 25 = 9

(x + 5)² = 9

x + 5 = ±3

x = ±3 - 5

x = -8 or -2

Interestingly, this problem is actually easier to solve by factoring:

x² + 10x + 16 = 0

(x + 8)(x + 2) = 0

x = -8 or -2

Answer 3