Question

How do you solve `p^2 + 3p - 9 = 0` by completing the square?

Answer 1

`p=-3/2-sqrt(45)/2` and `p=-3/2+sqrt(45)/2`

Explanation:

Step 1. Add and subtract the perfect square term.

The perfect square term starts with the value next to the variable `p`, in this case `3`. First, you cut it in half:

`3 => 3/2`

Then you square the result

`3/2 => (3/2)^2=9/4`

Then add and subtract this term in the original expression.

`p^2+3p+9/4-9/4-9=0`

Step 2. Factor the perfect square.

The terms in `color(blue)("blue")` are the perfect square.

`color(blue)(p^2+3p+9/4)-9/4-9=0`

`color(blue)((p+3/2)^2)-9/4-9=0`

Step 3. Simplify the remaining terms in `color(red)("red")`.

`(p+3/2)^2color(red)(-9/4-9)=0`

`(p+3/2)^2color(red)(-45/4)=0`

Step 4. Solve for `p`.

`(p+3/2)^2-45/4=0`

Add `45//4` to both sides

`(p+3/2)^2=45/4`

Take `+-` the square root of both sides.

`p+3/2=+-sqrt(45/4)`

Subtract `3//2` from both sides.

`p=-3/2+-sqrt(45)/2`

`p=-3/2-sqrt(45)/2` and `p=-3/2+sqrt(45)/2`

Answer 2

`p=-3/2+-[3sqrt(5)]/2`

Explanation:

`p^2+ 3p-9=0` => add 9 to both sides:
`p^2+3p=9` => using;`(a+b)^2=a^2+ 2ab+b^2`
`p^2+3p+(3/2)^2= 9 + 9/4`
`(p+3/2)^2=45/4` => take square root of both sides:
`p+3/2=+-sqrt(45)/2` => simplify:
`p=-3/2+-[3sqrt(5)]/2`