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

1 Answer
Jul 28, 2015

Answer:

#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)#