Skip main navigation

Al Khwarizmi’s Identity and the Quadratic Formula

The quadratic formula is a main tool for understanding quadratic relations. It is based on al Khwarizmi's technique of completing the square.
Picture of Al Khwarizmi
© UNSW Australia 2015
Al Khwarizmi is often considered the father of algebra, due to an influential text he wrote, and his name is the origin of the term algorithm. His `completing-the-square’ technique lies at the heart of a beautiful formula that we call al Khwarizmi’s identity. The usual quadratic formula is a consequence.
In this article you
  • will see how the completing-the-square leads to al Khwarizmi’s identity
  • how the quadratic formula follows from al Khwarizmi’s identity
  • use the quadratic formula to factor quadratic polynomials.
On this page is a very important derivation. Make sure you proceed slowly and carefully, please check all the steps by writing things out line by line, and then apply your understanding to work out the answers to the questions. This is how we learn mathematics.

Solving a quadratic equation using completing the square

Suppose we want to solve the quadratic equation
Half of the coefficient of (normalsize{x}) is (normalsize{-7}), so we take the (normalsize{1887}) to the other side, and add the square of (normalsize{-7}) to both sides. This gives
Now we rewrite the left-hand side as a perfect square:
At this stage we have to “take the square root” of (normalsize{1936}). What does this mean? It means finding a number (normalsize{r}) with the property that (normalsize{r^2=1936}). In this case such a number actually exists: it is (normalsize{r=44}). But otherwise we would just write (normalsize{pm sqrt{1936}}) to represent an approximate square root, and its negative. We can’t forget about the negative, since we want two solutions!
So in our case (normalsize{x-7=44}) or (normalsize{x-7=-44}). Thus we do get two solutions, namely (normalsize{x=51}) or (normalsize{x=-37}).

Deriving al Khwarizmi’s identity

Now let’s apply this to transform the general quadratic polynomial
where (aneq 0). First step, factor out the (normalsize{a}) to get
[Large{ax^2+bx+c = a left( x^2+frac{b}{a}x+frac{c}{a}right) }.]
Add and subtract (normalsize{left(frac{b}{2a}right)^2}) inside the right hand expression:
[Large{ax^2+bx+c = a left( x^2+frac{b}{a}x + left(frac{b}{2a}right)^2 +frac{c}{a} -left(frac{b}{2a}right)^2right)}.]
Now there is a perfect square inside these brackets:
[Large{ax^2+bx+c = a left( left(x+frac{b}{2a}right)^2 +frac{c}{a} -left(frac{b}{2a}right)^2right)}.]
[{Large frac{c}{a} -left(frac{b}{2a}right)^2 = frac{4ac-b^2}{(2a)^2}}]
we get
[Large{ax^2+bx+c = a left( left(x+frac{b}{2a}right)^2 +frac{4ac-b^2}{(2a)^2}right)}.]
Now for the last step, we multiply through by (normalsize{a}) to get
[Large{ax^2+bx+c=aleft(x+frac{b}{2a}right)^2+frac{4ac-b^2}{4a}. label{1} tag 1}]
This is one of the really great derivations in mathematics, resulting in an essential formula, which we call al Khwarizmi’s identity. This step has lots of questions to give you practice with this formula!
Q1 (E): Apply the steps above to rewrite (normalsize{x^2+12x-85}) using al Khwarizmi’s identity.
Q2 (M): Apply the steps above to rewrite (normalsize{3x^2+5x-22}) using al Khwarizmi’s identity.

Solving quadratic equations using al Khwarizmi’s identity

Once we have written a quadratic function in the form of al Khwarizmi’s identity, solving a quadratic equation involving it is relatively easy.
For example suppose that we want to solve (normalsize{x^2+12x-85=0}). Having already solved Q1, we rewrite this equation as
and then as
Now taking square roots, (normalsize{x+6=11}) or (normalsize{x+6=-11}) from which we deduce that (normalsize{x=5}) or (normalsize{x=-17}). These are the two solutions.
Q3 (M): i) Solve (normalsize{x^2+12x-85=-21}).
Q4 (M): i) Solve (normalsize{3x^2+5x-22=0}) using al Khwarizmi’s identity.

The quadratic formula

The familiar quadratic formula follows from al Khwarizmi’s identity.
If (normalsize{ax^2+bx+c = 0}) then
[Large{ left(x+frac{b}{2a}right)^2 = frac{b^2-4ac}{(2a)^2}}.]
Taking square roots
[Large{ x+frac{b}{2a}=pmfrac{sqrt{b^2-4ac}}{2a}}.]
And finally, we isolate (x)
[Large{ x= frac{- b pmsqrt{b^2-4ac}}{2a}}.]
This is the famous equation that all students memorise.
Q5 (E): Use the quadratic formula to solve (normalsize{x^2+4x+3=0}).

Factoring quadratics–the simpler way

Once we know how to solve quadratics, Descartes theorem allows us a simpler way to factor a quadratic expression: just find its zeroes first, and then each of these will correspond to a linear factor!
Q6 (E) Use the quadratic formula and Descartes’ theorem to factor (normalsize{x^2+x-132}).


A1. We transform the quadratic expression (normalsize{x^2+12x-85}) to
A2. We transform
We use the working of A1. to rewrite (normalsize{x^2+12x-85=-21}) as
This gives
In this case we can take square roots, giving (normalsize{x+6=10}) or (normalsize{x+6=-10}), from which we deduce that (normalsize{x=4}) or (normalsize{x=-16}). These are the two solutions.
We use the working of A2. to write (normalsize{3x^2+5x-22=0}) as
This gives
So taking square roots, (normalsize{x+5/6=17/6}) or (normalsize{x+5/6=-17/6}) from which we deduce that (normalsize{x=12/6=2}) or (normalsize{x=-22/6=-11/3}). These are the two solutions.
A5. Using the quadratic formula we have:
A6. Solving equation (normalsize{x^2+x-132=0}) we have zeroes
You are now in a position to systematically solve all quadratic factorisation problems, without any guess work. Thank you al Khwarizmi!
© UNSW Australia 2015
This article is from the free online

Maths for Humans: Linear and Quadratic Relations

Created by
FutureLearn - Learning For Life

Our purpose is to transform access to education.

We offer a diverse selection of courses from leading universities and cultural institutions from around the world. These are delivered one step at a time, and are accessible on mobile, tablet and desktop, so you can fit learning around your life.

We believe learning should be an enjoyable, social experience, so our courses offer the opportunity to discuss what you’re learning with others as you go, helping you make fresh discoveries and form new ideas.
You can unlock new opportunities with unlimited access to hundreds of online short courses for a year by subscribing to our Unlimited package. Build your knowledge with top universities and organisations.

Learn more about how FutureLearn is transforming access to education