
Standard Form
The standard form of the quadratic equation is a polynomial of the second degree, written as $ax\xb2+bx+c=0$,
where $x$ is unknown and $a,\; b,\; c$ are real numbers with $a\; \ne \; 0$. The equation is not quadratic
if $a=0$, because the term $ax\xb2$ is reduced to zero leaving only $bx+c=0$, which is the
equation of a line.
The Quadratic Formula
The easiest way to solve a quadratic equation is probably the quadratic formula. The quadratic formula can be derived by completing the
square on the quadratic equation $ax\xb2+bx+c=0$. The quadratic formula is:
$\backslash (\backslash textrm\{x=\}\backslash )$\(\frac{\textrm{b}\pm\sqrt{\textrm{b}^2\textrm{4ac}}}{\textrm{2a}}\)$(a\; \ne \; 0)$
Using the quadratic formula to solve a quadratic equation is straightforward, as shown by the following example:
Given the equation $x\xb2+10x+21=0$, we get:
$\backslash (\backslash textrm\{x=\}\backslash frac\{10\backslash pm\backslash sqrt\{10^24(1)(21)\}\}\{2(1)\}\backslash )$
$\backslash (\backslash ,\backslash ,\backslash textrm\{=\}\backslash frac\{10\backslash pm\backslash sqrt\{10084\}\}\{2\}\backslash )$
$\backslash (\backslash ,\backslash ,\backslash textrm\{=\}\backslash frac\{10\backslash pm\backslash sqrt\{16\}\}\{2\}\backslash )$
$\backslash (\backslash ,\backslash ,\backslash textrm\{=\}\backslash frac\{10\backslash pm\; 4\}\{2\}\backslash )$
$\backslash (\backslash ,\backslash ,\backslash textrm\{=\}\backslash \{\backslash frac\{104\}\{2\},\; \backslash frac\{10+4\}\{2\}\backslash \}\backslash )$
$\backslash (\backslash ,\backslash ,\backslash textrm\{=\}\backslash \{\backslash frac\{14\}\{2\},\; \backslash frac\{6\}\{2\}\backslash \}\backslash )$
$\backslash (\backslash ,\backslash ,\backslash textrm\{=\}\backslash \{\backslash )7,\; 3\backslash (\backslash \}\backslash )$
Solution by Factoring
Solving quadratic equations can be done using various methods. One way to solve a quadratic equation is by factoring. It is
important to keep in mind, however, that not every quadratic equation can be factored. Factoring a trinomial expression is part
of another topic (please refer to Factoring Polynomials for a detailed stepbystep
explanation.)
Once the quadratic equation is factored, the roots of the equation can be easily found. For example, the quadratic expression
$x\xb2+7x+12$ factors to $(x+4)(x+3)$. Since assigning a value of either $3\; or\; 4$ to $x$
satisifies the equation $x\xb2+7x+12=0$, the solution set is $x=\{4,3\}$.
A slightly more complicated example is one such as $6x\xb2+6x12$. After factoring, it becomes $(2x+4)(3x3)=0$.
We then need to solve each of the roots algebraically:
$2x+4=0\; \to \; 2x=4\; \to \; x=2;$
$3x3=0\; \to \; 3x=3\; \to \; x=1;$
The solution set is therefore $x=\{2,\; 1\}$.
Completing the Square
Since it's not possible to factor every quadratic expression, we often solve quadratic equations by the method known as completing
the square. In order to solve a quadratic equation by completing the square, we perform the following steps:
1) Write the equation on the standard form $ax\xb2+bx+c=0$.
2) Divide both sides of the equation by the coefficient of $x\xb2$ if not $1$ (i.e, divide by $a$.)
3) Subtract the constant term from both sides (i.e, subtract $c$.)
4) Divide the coefficient of $x$ by $2$, square the result, then add the resulting value to both sides (i.e, add $(\xbdb)\xb2$ or $\xbcb\xb2$.)
5) Factor the left side of the equation.
6) Apply the square root property $\pm \surd \dots $
7) Check the results if required. (i.e, plug the results into the equation to test it.)
8) Write the solution set.
For example, given the quadratic equation $3x\xb210x2$, we follow the steps given above:
1) Write the equation in standard form:
$3x\xb210x2=0$
2) Divide both sides of the equation by the coefficient of $x\xb2$, which is $3$:
$x\xb2\backslash (\backslash frac\{10\}\{3\}\backslash )x\backslash (\backslash frac\{2\}\{3\}\backslash )=0$
3) Subtract the constant term from both sides:
$x\xb2\backslash (\backslash frac\{10\}\{3\}\backslash )x=\backslash (\backslash frac\{2\}\{3\}\backslash )$
4) Divide the coefficient of $x$ by $2$, square the result, then add the resulting value to both sides:
$$(\(\frac{10}{3}\)⋅\(\frac{1}{2}\))^{²}=(\(\frac{5}{3}\))^{²}=\(\frac{25}{9}\)
$x\xb2\backslash (\backslash frac\{10\}\{3\}\backslash )+x+\backslash (\backslash frac\{25\}\{9\}\backslash )=\backslash (\backslash frac\{2\}\{3\}\backslash )+\backslash (\backslash frac\{25\}\{9\}\backslash )=\backslash (\backslash frac\{31\}\{9\}\backslash )$
5) Factor the left side of the equation:
$$($x\backslash (\backslash frac\{5\}\{3\}\backslash )$)^{²}=\(\frac{31}{9}\)
6) Apply the square root property:
$x\backslash (\backslash frac\{5\}\{3\}\backslash )=\; \pm \backslash (\backslash sqrt\{\backslash frac\{31\}\{9\}\}\backslash )$
$x\backslash (\backslash frac\{5\}\{3\}\backslash )=\; \pm \backslash (\backslash frac\{\backslash sqrt\{31\}\}\{3\}\backslash )$
$x=\backslash (\backslash frac\{5\}\{3\}\backslash )\pm \backslash (\backslash frac\{\backslash sqrt\{31\}\}\{3\}\backslash )$
7) Check the results if required:
$3$(\(\frac{5}{3}+\frac{\sqrt{31}}{3}\))^{²}$10$(\(\frac{5}{3}+\frac{\sqrt{31}}{3}\))$2=0$
$3$($\backslash (\backslash frac\{5\}\{3\}\backslash frac\{\backslash sqrt\{31\}\}\{3\}\backslash )$)^{²}$10$(\(\frac{5}{3}\frac{\sqrt{31}}{3}\))$2=0$
8) Write the solution set:
$x\; =${\(\frac{5\sqrt{31}}{3}\), \(\frac{5+\sqrt{31}}{3}\)} (or)
$x\; =${\(\frac{5±\sqrt{31}}{3}\)}
Since completing the square is more complex than factoring or using the quadratic formula, it's always a good idea to try to use those
methods before using this one (that is, unless you're doing homework and the assignment is to solve the problem by completing the square!)

