Question

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

Answer 1

See the entire solution process below:

Explanation:

First, subtract `color(red)(3x)` and `color(blue)(9)` from each side of the equation to put the quadratic equation in standard form:

`2x^2 - color(red)(3x) - color(blue)(9) = 3x + 9 - color(red)(3x) - color(blue)(9)`

`2x^2 - 3x - 9 = 3x - color(red)(3x) + 9 - color(blue)(9)`

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

`2x^2 - 3x - 9 = 0`

Next, factor the quadratic equation:

`(2x + 3)(x - 3) = 0`

Now, solve each term for `0`:

Solution 1)

`2x + 3 = 0`

`2x + 3 - color(red)(3) = 0 - color(red)(3)`

`2x + 0 = -3`

`2x = -3`

`(2x)/color(red)(2) = -3/color(red)(2)`

`(color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) = -3/2`

`x = -3/2`

Solution 2)

`x - 3 = 0`

`x - 3 + color(red)(3) = 0 + color(red)(3)`

`x - 0 = 3`

`x = 3`

`(2x)/color(red)(2) = -3/color(red)(2)`

The solution is: `x = -3/2` and `x = 3`