Question

How do you solve the quadratic equation by completing the square: `3x^2+9x-1=0`?

Answer 1

`x = -3/2 +-sqrt(31)/(2sqrt(3))`
`color(white)("XXXX")` `color(white)("XXXX")` `color(white)("XXXX")`(by completing the square)

Explanation:

Given `3x^2+9x-1=0`

Move the constant to the right side and extract the common factor of `3` from the left side:
`color(white)("XXXX")` `3(x^2+3x) = 1`

If `x^2+3x` are the first two terms of the expansion of a squared binomial:
`color(white)("XXXX")` `(x+a)^2 = (x^2+2ax+a^2)`

then `a= 3/2` and `a^2 = (3/2)^2 = 9/4`
`color(white)("XXXX")`which is the amount we will need to add (inside the parentheses) to "complete the square)

`color(white)("XXXX")`Note that adding `9/4` inside the parentheses is the same as adding `3*(9/4)` and this amount will need to be added to the right side as well to keep the equation balanced.

`color(white)("XXXX")` `3(x^2+3x+(3/2)^2) = 1 + 27/4`

Rewriting as a squared binomial (and simplifying the right side)
`color(white)("XXXX")` `3(x+3/2)^2 =31/4`

Dividing both sides by `3`
`color(white)("XXXX")` `(x+3/2)^2 = 31/12`

Taking the square root of both sides:
`color(white)("XXXX")` `x+3/2 = +- sqrt(31/12)`

Subtracting `3` from both sides
`color(white)("XXXX")` `x = -3/2 +-sqrt(31)/(2sqrt(3))`