# What are the benefits of using $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ to solve quadratic equations?

When I was in high school, they taught me to solve quadratic equations with this formula:

$$x=\frac{\sqrt{4 \text{ac}+b^2}-b}{2 a}$$

EDIT: The original formula is this one: $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$, I confused it with another formula on Wikipedia.

While reading Mathematics for the Nomathematician, Morris Kline suggests that a quadratic equation should be solved by making another equation with half of the coefficient of $b$:

$$y=x+\frac{b}{2}$$

Then:

$$x=y-\frac{b}{2}$$

and then proceed with the substitution on the quadratic equation, I guess I don't need to describe the rest as it may be a standard procedure.

Are there benefits on using the first formula? I also imagine that they could be related somehow, but I still can't see this relation.

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Unless your quadratics are different from everyone else's, don't you want $-4ac+b^2$ under that square root? –  Gerry Myerson Aug 3 '12 at 23:13
Assuming $a=1$, Kline is describing "completing the square". –  Robert Israel Aug 3 '12 at 23:14
@Robert: Ah - this all suddenly makes sense to me. –  mixedmath Aug 3 '12 at 23:14
@mixedmath Yes. I missed something. –  Vladimir Putin Aug 3 '12 at 23:16
@mixedmath This was also one of the suggetions on wikipedia, I'm just not sure if that's right. –  Vladimir Putin Aug 3 '12 at 23:17

It's the same thing. What Kline is doing is what's in general known as a depression, which is a variable substitution done for a polynomial of degree $n$, such that the resulting polynomial has no term of degree $x^{n-1}$.

One way to see how depression works is to consider the Vieta formulae, which in the case of the quadratic $ax^2+bx+c=a(x-x_1)(x-x_2)$ looks like this:

\begin{align*} x_1 x_2 &= \frac{c}{a}\\ x_1 + x_2 &= -\frac{b}{a} \end{align*}

From this, if the sum of the roots is $-\dfrac{b}{a}$, then the mean of the roots is $-\dfrac{b}{2a}$. Thus, the depression substitution

$$y=x+\frac{b}{2a}$$

can be geometrically interpreted as shifting the parabola corresponding to your quadratic $ax^2+bx+c$ such that it is "centered" about the origin, and the two roots are laid out symmetrically.

If we depress our original quadratic, we get

\require{cancel}\begin{align*} ax^2+bx+c&=a\left(y-\frac{b}{2a}\right)^2+b\left(y-\frac{b}{2a}\right)+c\\ &=a\left(y^2-\frac{b}{a}y+\frac{b^2}{4a^2}\right)+b\left(y-\frac{b}{2a}\right)+c\\ &=ay^2\cancel{-by}+\frac{b^2}{4a}\cancel{+by}-\frac{b^2}{2a}+c\\ &=ay^2+\frac{b^2}{4a}-\frac{2b^2}{4a}+c\\ &=ay^2-\frac{b^2}{4a}+c=ay^2-\frac{b^2-4ac}{4a} \end{align*}

and I'm sure you know how easy it is to solve the equation

$$ay^2-\frac{b^2-4ac}{4a}=0$$

Having solved for $y$, you undo the depression you did, which means you have to add the term $-\dfrac{b}{2a}$ to get the actual roots you want. That's where that part of the quadratic formula comes from.

In short, I would not say that Kline's method is the most expedient, but it at least looks to me that this slower method allows for more cogitation on what the steps are supposed to "mean", as opposed to a lazy plug-and-chug.

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@Gustavo, the search term you should be using is polynomial depression; you'll see that the general idea is also used in deriving explicit solutions for the cubic and quartic equations. –  Ｊ. Ｍ. Aug 4 '12 at 4:47
@Gustavo, depression can be done on any polynomial, but you should know that the general formulae for expressing the roots of polynomials of degree five and higher require "special functions"; Abel and others have shown that there are no general formulae for those roots that only involve elementary functions. –  Ｊ. Ｍ. Aug 4 '12 at 5:25
Well, things are slightly simpler if the polynomial is monic (that is, the leading coefficient is $1$), and when solving a quadratic, one can always perform a division so that you are working with a monic polynomial. –  Ｊ. Ｍ. Aug 4 '12 at 16:11
@Gustavo, congratulations; you've just formed a monic quadratic equation from the general form. –  Ｊ. Ｍ. Aug 4 '12 at 16:16
I'm also happy with something I just read: When doing mathematics, Stubbornness is more important than inteligence. It's cool because it's the only thing I have. Haha –  Vladimir Putin Aug 6 '12 at 4:41

Let's try to follow Kline's advice (and all the time we assume the coefficients $\,a,b,c\,$ are real).

We have $\,ax^2+bx+c=0\,\,,\,a\neq 0\,$ and we make the substitution $$y:=x+\frac{b}{2}\Longleftrightarrow x = y-\frac{b}{2}$$ and from here, substituting in the equation

$$a\left(y-\frac{b}{2}\right)^2+b\left(y-\frac{b}{2}\right)+c=0$$ ...and now? But for messing up the equation's variable I can't see any real advance or simplification.

I think what he actually meant, or should have meant, is the following, which is only the rather well-known method of completing the square (CS): $$X^2\pm BX=\left(X \pm \frac{B}{2}\right)^2-\frac{B^2}{4}\,\,,\,\text{and from here}$$ $$0=ax^2+bx+c\,\Longrightarrow\,x^2+\frac{bx}{a}=-\frac{c}{a}\stackrel{CS}\Longrightarrow\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a^2}=-\frac{c}{a}\,\Longrightarrow$$ $$\Longrightarrow\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}\,\,\,\text{(see the numerator?!)}\,\,\stackrel{\text{sq.rt. in both sides}}\Longrightarrow$$ $$\Longrightarrow x_{1,2}+\frac{b}{2a}=\pm \frac{\sqrt{b^2-4ac}}{2a}\,\Longrightarrow\,x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ .

So, as you can see, the benefits is that you need that expression to solve quadratics..as simple as that!

Of course, the number $\,\Delta:=b^2-4ac\,$ , called the quadratic's discriminant serves to know beforehand about the above equation's possible solutions: \begin{align*}\Delta>0\Longrightarrow & \,\text{there exist two different real solutions to the equation}\\ \Delta=0\Longrightarrow & \,\text{there exists only one unique real solution to the equation}\\ \Delta<0\Longrightarrow & \,\text{the equation has no real solutions}\end{align*}

In the last case above there exist two conjugate complex non-real solutions (disregard this if you haven't yet studied complex numbers)

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What's the meaning of $:=$? It's the first time I see it. –  Vladimir Putin Aug 4 '12 at 16:16
:= means the left side is defined by the right one –  DonAntonio Aug 11 '12 at 1:16
"there exists only one unique real solution to the equation" - slightly strange wording, like, there are actually two solutions, but the other one isn't unique? :) –  Ben Millwood Aug 11 '12 at 10:23
@BenMillwood, I think that's a language fix. Where I live (Israel), even at 7th-8th grade in junion high school, children are taught to say "there are (there aren't) REAL solution(s) to this quadratic equation", apparently as preparation for complex numbers later in high school. The meaning, though, is correct, I believe, under the proper interpretation: one unique real solution means there's only one number that solves the quadratic eq. –  DonAntonio Aug 11 '12 at 10:37
@DonAntonio: I see where you're coming from, but I think it's strange to use "unique" here - what are you saying that isn't said by "there exists only one real solution to the equation"? If you think the latter is untrue because there is a repeated solution, then you think there are two solutions, but then you say that there is one unique solution, which just leads me to wonder if the other one is unique, too... –  Ben Millwood Aug 11 '12 at 10:47