Do you have to complete the square before using the quadratic formula?

Question

so lets say we have $x^2+4x-21$. We can solve this by completing the square (divide, square, add I know the drill). But do we have to do this when we use the quadratic formula? And if not, why would we ever need to complete the square?

Thanks!

Answer 1

No, you don't need to complete the square (or do any other thing) when using the quadratic formula except substitute the values of the coefficients of the unknown powers.

As to why you need to complete the square, note that first, the quadratic formula can be derived by completing the square on the general quadratic equation. Secondly, in mathematics we enjoy having different methods of performing the same effective operation because some are more efficient than others in particular situations. Note that sometimes when you can quickly factorise a quadratic, it is unnecessary to use other means as that would require additional computation. Similarly, there are some times when one can easily complete the square; in such cases using the general formula is just a waste of effort and time. So it is good to sometimes know more than one way to do something.

All in all though, it doesn't matter for most digital computers, most of which are programmed to evaluate the general formula in the form you know to solve quadratic equations.

Answer 2

In fact the quadratic formula was found out by using the completion of squares method!

We have: $$ax^2+bx+c=0$$ $$a\left[x^2 +\frac{b} {a} x + \frac{c} {a} \right] =0$$ $$a\left[\left(x + \frac{b} {2a}\right)^2 + \frac{c}{a} - \frac{b^2}{4a}\right]=0$$ $$x = - \frac{b} {2a} \pm \frac{\sqrt{b^2-4ac}}{2a}$$

Answer 3

As Rohan has already pointed out, there is an explicit formula for finding the real-valued roots (if any such roots exist) of any quadratic equation $ax^2 + b+x +c = 0$. So the answer is no: To solve a quadratic equation, we do not need to complete the square. We can simply apply the general formula. But the general formula for the roots of a quadratic equation was obtained by means of the method of completing a square.

Answer 4

You don’t need to complete the square first. Just identify the “a, b, c” by looking at the coefficients of the equation you have and plug them into the quadratic formula.

Answer 5

If all you want to do is solve simple quadratic equations like $x^2+4x-21 = 0$, you can just use the quadratic formula—which is exactly what you get if you complete the square ahead of time, for the general equation, and then plug in the coefficients as late as possible.

But the technique of completing the square is useful in more general contexts, not just for discovering the quadratic formula. For example, suppose you have the equation $$x^2+y^2 = 3x$$ and you are wondering what it will look like when you graph it. You might already know that $x^2+y^2 = r^2$ is a circle with radius $r$ and center at $(0,0)$, and this looks something like that, but not exactly. By completing the square on the original equation, you can put it in that form and see that it is a circle:

$$\begin{align} x^2 && + y^2 & = 3x \\ x^2 - 3x & & + y^2 & = 0 \\ \color{maroon}{x^2 - 3x}&\color{maroon}{ + \left(\frac32\right)^2} \color{darkblue}{- \left(\frac32\right)^2} & + y^2 & = 0 \\ \color{maroon}{\left(x - \frac32\right)^2} && + y^2 & = \color{darkblue}{\left(\frac32\right)^2} \end{align} $$

and now it is in the right form, but with the center at $\left(x-\frac32,y\right) = (0,0)$ instead of at $(x,y) = (0,0)$. And you can read off the radius, which is $\frac32$. Without completing the square this would not be at all obvious.

Or to take a more advanced example, suppose you're trying to calculate $$\int\frac{dx}{x^2-2x}.$$ Completing the square in the denominator allows you to rewrite the integrand as $$\int\frac{dx}{(x-1)^2 - 1}$$ and then you make the substitution $y=x-1$: $$\int\frac{dy}{y^2-1}$$ which can be handled with partial fraction methods.

Being able to rewrite $x^2+bx$ as $y^2+c$ is frequently convenient.