Does completing the square always work for quadratic equations?

Created by Anna Szczepanek, PhD
Reviewed by Rijk de Wet
Last updated: Jan 27, 2022

Completing the square is a method to solve quadratic equations that always works! We will explain the logic behind this method and we'll show you an example of how it works. Also, we discuss the advantages of this method over the quadratic formula and vice versa.

What is completing the square and when do you use it?

Completing the square is an algebraic technique that lets you solve quadratic equations (or equivalently, factor quadratic trinomials). With this method, you apply basic algebraic operations to transform your problem so that there's a perfect square trinomial on the left side of the equation and a constant term on the right side. Formally, for the equation x2+bx+c=0x^2 +bx + c = 0 we perform the following transformations:

  1. Add b24c\frac{b^2}{4} - c to both sides:

    x2+bx+c+(b24c)=b24cx^2 + bx + c + (\frac{b^2}{4} - c) = \frac{b^2}{4} - c
    x2+bx+b24=b24c\therefore x^2 + bx + \frac{b^2}{4} = \frac{b^2}{4} - c

  2. Use the short multiplication formula in reverse to transform the left side:

    x2+bx+b24=(x2+b2)2x^2 + bx + \frac{b^2}{4} = (x^2 + \frac{b}{2})^2
    (x2+b2)2=b24c\therefore (x^2 + \frac{b}{2})^2 = \frac{b^2}{4} - c

  3. Look at b24c\frac{b^2}{4} - c on the right side.

    • If it's positive, we take the square root of both sides. The equation then has two solutions.
    • If it's equal to zero, we immediately know that the solution is x=b2x = -\frac{b}{2}.
    • If it's negative, then our equation has no real solutions and only complex solutions.

The method of completing the square will work for every quadratic equation you can imagine. Above, we assumed that the coefficient for x2x^2 is equal to 11. If your equation starts with ax2ax^2 and aa is not 11 (for example, if you need to solve 2x2+3x7=02x^2+3x-7=0), then divide both sides of the equation by aa (in this example it's 22). If done correctly, the left side will now start with x2x^2, and you can then apply the steps we've discussed above.

Does completing the square always work?

Yes, you can solve any quadratic equation by completing the square, even if the equation has no real solutions! However, in some cases, quite a bit of computation will be required. Here's an example of an equation without real roots solved by completing the square:

"Solve x24x+13=0x^2-4x+13 = 0."

  1. Subtract 99 from both sides:

    x24x+139=9x^2-4x+13-9 = -9
    x24x+4=9x^2-4x+4 = -9

  2. Recognize the perfect square on the left-hand side:

    (x2)2=9=(3i)2(x-2)^2 = -9 = (3i)^2

  3. Take the square root:

    x2=9=±3ix-2 = \sqrt{-9} = \pm 3i

  4. The final answer is:

    x=2±3ix = 2 \pm 3i

Why does completing the square work?

The method of solving quadratic equations by completing the square always works because of the short multiplications formula:

(x+b)2=x2+bx+b2(x+b)^2 = x^2+bx+b^2

Consequently, for every quadratic expression of the form x2+bx+cx^2+bx+c you can add (or subtract) a constant term on both sides of the equation so that we obtain the perfect square trinomial (x+b)2(x+b)^2.

What is the benefit of completing the square?

It is certainly a good idea to learn the completing the square method. First, it is much more intuitive than simply plugging the coefficients into the quadratic formula. It gives you a feeling of understanding what happens to your equation and why a particular number is a solution. In fact, it's almost impossible to forget how this method works once you've seen it in action!

Also, the method of completing the square can be used to derive the quadratic formula, should you ever forget it — and it's easy to forget a formula that you simply learned by heart and never fully understood.

Finally, if you ever face integration problems, you'll be thankful you've mastered completing the square, because it pops up repeatedly there.

Is there ever an advantage to using the quadratic formula?

The advantage of using the quadratic formula method for solving quadratic equations is that it usually requires fewer steps and takes less time than completing the square.

Moreover, it can be easily implemented if you need to solve quadratic equations inside a computer program.

Anna Szczepanek, PhD
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