Quadratic equation

In mathematics, or more specifically algebra, a quadratic equation is an equation involving only polynomials of the second degree. Quadratic equations are a common part of mathematical solutions to real-world problems in a huge variety of situations. Fortunately, there exists a simple closed formula for finding the roots of such an equation, the quadratic formula, involving no operation more advanced than taking the square root of a real number.

Every polynomial equation can be put into the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2+bx+c=0\,}

with a, b and c real and $a\not =0$. The quadratic formula specifies the roots of this equation as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\ .}

Here, there are actually two roots being given, although the notation obscures this somewhat. One root is obtained by replacing the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm} appearing in the formula by a + sign, and the other is obtained by replacing it with a - sign. The formula is guaranteed to work for all quadratic polynomials, but sometimes the roots will be complex numbers even when every other part of the problem deals only with real numbers. Looking at the quadratic formula, we see that the roots will be complex precisely when the discriminant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta = b^2 - 4ac } is negative.

There are other less frequently used techniques for solving quadratic equations. Completing the square is just as general as the quadratic formula (in fact, it is used to derive the formula below). Otherwise, if the polynomial has no linear term, or if it is easy to factor, there are faster methods than using the quadratic formula.

In this article, we are assuming as above that the coefficients of the quadratic polynomial are real. Quadratic equations can occur involving many other types coefficients, and the quadratic formula, interpreted correctly, can be applied in many of these other situations as well. See the advanced subpage for information about these more general quadratic equations.

The problem

Any real second-degree polynomial in the variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} will be of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2+bx+c\,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} , $b$, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} are real constants and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is not zero (if it was, the polynomial would only be first-degree). The graph of the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = ax^2 + bx + c} a parabola, and the roots that the quadratic equation will give us are the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} at which the parabola crosses the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis. This means that the roots of the polynomial are the particular values of $x$ for which the polynomial equals zero.

The problem that the quadratic formula solves is to find those roots.

The solution

The Fundamental Theorem of Algebra tells us that we should expect there to be two roots for a second-degree polynomial, although they might be equal in some cases, and may not be real even when the coefficients and variable are. If we call the roots Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_+} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_-} then what we are saying is that we have the quadratic equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax_\pm^2+bx_\pm+c=0\ .}

This is where the quadratic formula comes in. It tells us that the solutions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_+} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_-} can always be found as

$x_{\pm }={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\ .$

Looking at the above result, it is clear that the discriminant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta= b^2-4ac} is of interest for two reasons. First it is the part of the solution that is either positive or negative depending on whether you are looking at the root Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_+} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_-} , so it determines whether there are two different roots or just one. Second it is under a square root, so we must wonder what happens when it is negative. This gives us three cases to look at.

Figure 1: A parabola with two real roots. This corresponds to a second-degree (real) polynomial with a positive discriminant.

Figure 2: A parabola with two only one real root, but of multiplicity 2. This corresponds to a second-degree (real) polynomial with a discriminant equal to zero.

Figure 3: A parabola with no real roots, as seen by the fact that it does not intersect the horizontal axis. This corresponds to a second-degree (real) polynomial with a negative discriminant.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta>0}

Here we have two distinct real roots, since the square root of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta} will also be real and greater than zero, meaning that we have

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

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_-=x_\pm=\frac{-b-\sqrt{b^2-4ac}}{2a}\ .}

These can be seen graphically as the two red dots in Figure 1. In this case we can rewrite the polynomial in terms of its roots as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2+bx+c = \left(x-x_+\right)\left(x-x_-\right)\ ,}

and it is easy to see that indeed if we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x_+} or $x=x_{-}$ then the polynomial well be equal to zero.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta=0}

Now there is only one distinct root. It is still real, and is said to have a multiplicity of 2. This is because the root is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0=\frac{-b}{2a}}

and the polynomial can be expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2+bx+c = \left(x-x_0\right)^2\ .}

This case occurs when the parabola described by the polynomial just touches the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis at exactly one point, the red dot shown in Figure 2. Again, setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x_0} in the polynomial makes it vanish.

$\Delta <0$

For negative values of the discriminant there are no real roots to the polynomial. Graphically this corresponds to the situation where the lowest point on the parabola is above the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis (or the highest point is below it, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is negative) as shown in Figure 3. In this case the roots still exist, as guaranteed by the fundamental theorem of algebra, but they are complex so cannot be shown on the real number line.

Proof

The simplest way to show that the values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_\pm} are in fact roots to the polynomial above is to substitute them into the equation

${\begin{aligned}ax_{\pm }^{2}+bx_{\pm }+c&=a\left({\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\right)^{2}+b{\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}+c\\&={\frac {1}{4a}}\left(b^{2}\mp 2b{\sqrt {b^{2}-4ac}}+b^{2}-4ac\right)-{\frac {b^{2}\mp b{\sqrt {b^{2}-4ac}}}{2a}}+c\\&={\frac {b^{2}\mp b{\sqrt {b^{2}-4ac}}}{2a}}-c-{\frac {b^{2}\mp b{\sqrt {b^{2}-4ac}}}{2a}}+c\\&=0\ ,\end{aligned}}$

as desired. Notice that this proof is valid even in the case where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta} is less than zero.

Derivation of the quadratic formula

To derive the quadratic formula stated above, we must start with the quadratic equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2+bx+c=0\,}

and then complete the square. We begin by subtracting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} from and then multiplying both sides by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4a} , to obtain

$4a^{2}x^{2}+4abx=-4ac.\,$

As written, the left side does not factor in a useful way. The trick is to add Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2} to both sides, because afterward the left side will factor as a complete square:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(2ax+b \right)^2= b^2-4ac .\,}

We now take the square root of both sides, remember that there will be two possible square roots for of the number on the right side, indicated by the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm} symbol:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2ax + b = \pm \sqrt{b^2-4ac}. }

Finally, we solve for $x$ to obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}} ,

which is the quadratic formula.