How do you solve the quadratic equation by completing the square: 5x^2 + 8x - 2 = 0?

1 Answer
Aug 5, 2015

x_(1,2) = -4/5 +- sqrt(26)/5

Explanation:

To solve this quadratic by completing the square, you first need to get your quadratic to the form

color(blue)(x^2 + b/a = -c/a)

In your case, this means adding 2 to both sides of the equation and dividing everything by 5

5x^2 + 8x - color(red)cancelcolor(black)(2) + color(red)cancelcolor(black)(2) = 0 + 2

(color(red)cancelcolor(black)(5)x^2)/color(red)cancelcolor(black)(5) + 8/5x = 2/5

x^2 + 8/5x = 2/5

Next, use the coefficient of the x-term to help you determine what term needs to be added to both sides of the equation so that the left side can be written as the square of a binomial.

More specifically, divide said coefficient by 2, square the result, then add it to both sides of the equation

(8/5 * 1/2)^2 = 16/25

Your equation will now look like this

x^2 + 8/5x + 16/25 = 2/5 + 16/25

The left side of the equation can be written as

x^2 + 2 * (4/5) * x + (4/5)^2 = (x + 4/5)^2

This means that you now have

(x + 4/5)^2 = 26/25

Take the square root of both sides to get

sqrt( (x+4/5)^2) = sqrt(26/25)

x + 4/5 = +- sqrt(26)/5 implies x_(1,2) = -4/5 +- sqrt(26)/5

The two solutions to your quadratic will be

x_1 = color(green)(-4/5 - sqrt(26)/5) and x_2 = color(green)(-4/5 + sqrt(26)/5)