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 Skip Navigation LinksMath Help > Basic Algebra > Factoring > Factoring: The Quadratic Formula

This is the third page about factoring.   It helps you FACTOR QUADRATIC EQUATIONS

In the first page, I presented the perfect square trinomial and the difference of two squares.

In the second page, I presented a method of factoring four-term expressions with pairs of factors.

In the third, group of pages, I presented methods of factoring trinomials.

Here, I will present the QUADRATIC FORMULA.  You may be interested to know the quadratic formula can be derived fairly easily using the concepts of the perfect square trinomial and the difference of two squares.

How to recognize QUADRATIC EQUATIONS
It has the following form:

ax² + bx + c = 0, where a is not zero.


Before I show you a method of deriving the quadratic formula, would you like to see a page about finding two numbers if you know their sum and their product?  This topic is explored in great detail there, along with a little javascript calculator that finds these numbers, along with a complete explanation of how it works.


Here on this page, I will show you a method of deriving the quadratic formula, which is to change the general form of the quadratic equation into an equivalent form that can be factored, assuming b² - 4ac ≥ 0:

ax² + bx + c = 0, so
x² +   b  x +   c   = 0,  dividing through by a, which is legal because a 0.
 a  a
x² +   b  x +   b²   -   b²   +   c   = 0.   I just added and subtracted the same value to make a perfect square trinomial.
 a  4a²  4a²  a
(x +   b  )² -   b²   +   c   = 0.   I factored the perfect square trinomial.
 2a  4a²  a
(x +   b  )² - (  b²   -   c  ) = 0.   Collected terms.
 2a  4a²  a
(x +   b  )² - (  b² - 4ac ) = 0.   Found a common denominator.
 2a     4a²
(x +   b  )² - ( )² = 0.   Because u = ( u )², as long as u ≥ 0.  Now we have the difference of two squares.
b² - 4ac
 2a
      2a       
(x +   b   +  )  ( x +   b   -  ) = 0.  
b² - 4ac b² - 4ac
 2a  2a
      2a              2a       
( x +  b + )  ( x +  b - ) = 0.  
b² - 4ac b² - 4ac
         2a                    2a          
Now we have derived a factored form of ax² + bx + c, which helps us find roots of the equation, the values of x for which this trinomial is zero.  Since anything multiplied by zero is zero, then any value of x that makes either factor zero in the expression, above, will make the whole expression zero.  So now we have...
x =  -b -   or    x =  -b +
b² - 4ac b² - 4ac
         2a                    2a          
Finally, these two solutions are summarized in the following shorthand:
x =  -b �
b² - 4ac
         2a          

This quadratic formula gives us some interesting insights into the form of the graph of the equation,

y = ax² + bx + c

Shape of the graph

The graph has the shape of a parabola, which is the trajectory a ball follows when it is thrown.  If a is positive, the parabola opens upwards.  If a is negative, the parabola opens downwards.

Line of Symmetry

The parabola is symmetric about the line x = -b/(2a), which as you can see, is the average of the two roots (values of x for which the graph of the equation passes through the x-axis), if it has any.

Number of Roots (Solutions)

There are two of roots whenever b² > 4ac.  There is one root (or some people say two identical roots) whenever b² = 4ac, and there are no roots if b² < 4ac.

Vertex

The vertex of the parabolic graph of the equation is the point (h,k) on the line of symmetry through which the parabola passes.  Since it's on the line of symmetry, which is x=-b/(2a), we know that h=-b/(2a).  To find the value of k, simply substitute

h = -b/(2a)

in the equation

k = ax² + bx + c

And then solve for k.

k = a(-b/(2a))² + b(-b/(2a)) + c

k = b²/(4a) - b²/(2a) + c

k = b²/(4a) - 2b²/(4a) + 4ac/(4a)

k = -(b² - 4ac)/(4a)

There's that b² - 4ac again!  If a is positive (parabola opens up) and it has two roots (b² > 4ac) then the y value of the vertex is in negative territory, so you can see that both arms of the graph of the parabola pass through the x axis.  Similarly, if a is negative, and b² > 4ac then the vertex has a positive y coordinate, and the downward-opening parabola's arms pass through the x axis.  But if b² < 4ac, then the vertex is on the same side of the x axis as the direction of opening, so the parabola does not pass through the x axis.  Graph a few of these possibilities to convince yourself of these facts.

The vertex form of the quadratic equation y=ax²+bx+c is

y-k = a(x-h)²

where 

h = -b/(2a)

and

k = -(b² - 4ac)/(4a)

Alternative Derivation of Vertex Form

Here's another (quicker) way to derive the vertex form of the quadratic equation.

Start with the quadratic formula: y = ax² + bx + c
Subtract c from both sides: y - c = ax² + bx
Add b²/(4a) to both sides: y + b²/(4a) - c = ax² + bx + b²/(4a)
Factor "a" out of the right side: y + b²/(4a) - (4ac)/(4a) = a(x² + (b/a)(x) + b²/(4a²))
Factor the right side: y + b²/(4a) - (4ac)/(4a) = a(x + b/(2a))²
4a is the common denominator: y + (b² - 4ac)/(4a) = a(x + b/(2a))²
Now let k=-(b² - 4ac)/(4a)
and h=-b/(2a)
y - k = a(x - h)²

Focus and Directrix

Here's another interesting property of a parabola:  It has a "focus", which is a point, and a "directrix", which is a line, such that every point of the parabola is equidistant from the focus and the directrix.

Using the vertex form of the parabola,

y - k = a(x - h)²

where the point (h,k) is the vertex, we define the focus to be the point (h, k + 1/(4a)) and the directrix to be the line y = k - 1/(4a).  Let me emphasize this:

focus is (h, k + 1/(4a))

directrix is y = k - 1/(4a)

Now let's show that every point that satisfies y - k = a(x - h)² is equidistant from the focus and directrix.

First, let's find the distance from the focus to a point (x,y) on the parabola.  Since it's on the parabola, point (x,y)  satisfies y - k = a(x - h)².  Therefore, this point can be expressed (x, a(x - h)²+k)

The distance from the focus (h, k + 1/(4a)) to that point (x, a(x - h)²+k) is given by

d2 = (x-h)2 + (a(x-h)2 - 1/(4a))2

d2 = (x-h)2 + a2(x-h)4 - (x-h)2/2 + 1/(4a)2

d2 = a2(x-h)4 + (x-h)2/2 + 1/(4a)2

d2 = (a(x-h)2 + 1/(4a))2

d = a(x-h)2 + 1/(4a)

Now, the distance from the directrix y = k - 1/(4a) to the point (x, a(x - h)²+k) is the difference in the "y" values, a(x-h)2 + 1/(4a).  This distance from the directrix to the point of the parabola is the same as the distance from the focus to that same point.

An aside

The proof, above, of the quadratic formula isn't one that you might just conjure up on first thought.  In fact, it seems rather arbitrary that certain expressions were added to both sides of the equation.  Your math teacher might say, "well, I'm completing the square", but the easier way to remember and repeat the proof (in case you're ever asked to do that) is to write the result in factored form (the second-to-last line of the proof), and then derive each line of the proof from the one just below it -- in other words, work backwards from the result.

I used this same technique to show certain hyperbolic equivalences.

Related pages in this website

Cubic formula

Conic Sections

hyperbolic equivalences

The Cauchy-Schwarz Inequality, which is proved using the quadratic formula

Hyperbolic Functions -- sinh(x) and cosh(x)


The webmaster and author of this Math Help site is Graeme McRae.