How do you solve the quadratic equation by completing the square: v^2 - 2v = 3v2−2v=3?

1 Answer
Stefan V.
Jul 28, 2015

v_1 = 3v1=3, v_2 = -1v2=−1.

Explanation:

Your starting quadratic equation looks like this

v^2 color(blue)(- 2)v = 3v2−2v=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 xx-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 = -1−22=−1, then

(-1)^2 = +1(−1)2=+1

The quadratic becomes

v^2 - 2v + 1 = 3 + 1v2−2v+1=3+1

v^2 - 2v + 1 = 4v2−2v+1=4

The left side of the equation is equivalent to

v^2 - 2v + 1 = (v-1)^2v2−2v+1=(v−1)2

You thus have

(v-1)^2 = 4(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=√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)

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