Question

How do you solve the quadratic equation by completing the square: `v^2 - 2v = 3`?

Answer 1

`v_1 = 3`, `v_2 = -1`.

Explanation:

Your starting quadratic equation looks like this

`v^2 color(blue)(- 2)v = 3`

Now, to solve quadratic equations by completing the square you need to add a term to both sides of the equation such that the left side of the equation becomes the aquare of a binomial.

To do that, divide the coefficient of the `x`-term by 2 while keeping the sign, square it, and add the result to both sides of the equation.

In your case, you have

`(color(blue)(-2))/2 = -1`, then

`(-1)^2 = +1`

The quadratic becomes

`v^2 - 2v + 1 = 3 + 1`

`v^2 - 2v + 1 = 4`

The left side of the equation is equivalent to

`v^2 - 2v + 1 = (v-1)^2`

You thus have

`(v-1)^2 = 4`

To solve this equation, take the square root from both sides of the equation

`sqrt((v-1)^2) = sqrt(4)`

`v-1 = +-2 => v_(1,2) = +- 2 + 1 = {(v_1 = +2+1 = 3), (v_2 = -2 + 1 = -1) :}`

The two solutions to your equation are

`v_1 = color(green)(3)` and `v_2 = color(green)(-1)`