How Do I Complete the Square to Solve Quadratic Equations?

Another method that can be used to solve quadratic equations for their roots is called completing the square. This method is also used to convert quadratics in polynomial form to quadratics in vertex form, which I’ll discuss later.

If you’ve ever attempted to learn this method I’m sure you’ve heard “half it, square it, add it, it’s easy.” That quote does not adequately explain this process at all in my experience.

Consider the quadratic equation


The first thing to note is that this quadratic in polynomial form is a monic polynomial. This is to say that the leading coefficient is 1; in this case I’m referring to the coefficient on the x^2 term. Completing the square only works when the coefficient on the x^2 term is 1.

Already having a monic polynomial, the first step is to rewrite the equation with the constant term on the right hand side. In our case,


The next step is to take the coefficient from the x^1 term, and “half it, square it, and add it” to both sides. The revised equation should read


The next step is to rewrite the equation in square form. In our case


The sign between


remains positive because the coefficient on the x^1 term in the original quadratic is positive.

The next step is to take the square root of both sides of the equation. The revised equation should read


which leads to the final step which is solving for x. In our case



x=-(6/2) and x=-(4/2) or x=-3 and x=-2.


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