Skip main navigation
We use cookies to give you a better experience, if that’s ok you can close this message and carry on browsing. For more info read our cookies policy.
We use cookies to give you a better experience. Carry on browsing if you're happy with this, or read our cookies policy for more information.
Picture of Al Khwarizmi
Al Khwarizmi

Al Khwarizmi's identity and the quadratic formula

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 step 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.

This step is probably the algebraic highlight of the course, and the most detailed steps. 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 is , so we take the to the other side, and add the square of 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 . What does this mean? It means finding a number with the property that . In this case such a number actually exists: it is . But otherwise we would just write 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 or . Thus we do get two solutions, namely or .

Deriving al Khwarizmi’s identity

Now let’s apply this to transform the general quadratic polynomial

where . First step, factor out the to get

Add and subtract inside the right hand expression:

Now there is a perfect square inside these brackets:

Since

we get

Now for the last step, we multiply through by to get

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 using al Khwarizmi’s identity.

Q2 (M): Apply the steps above to rewrite 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 . Having already solved Q1, we rewrite this equation as

and then as

Now taking square roots, or from which we deduce that or . These are the two solutions.

Q3 (M): i) Solve .

Q4 (M): i) Solve using al Khwarizmi’s identity.

The quadratic formula

The familiar quadratic formula follows from al Khwarizmi’s identity.

If then

So

Taking square roots

And finally, we isolate

This is the famous equation that all students memorise.

Q5 (E): Use the quadratic formula to solve .

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 .

Answers

A1. We transform the quadratic expression to

A2. We transform

A3.

We use the working of A1. to rewrite as

This gives

In this case we can take square roots, giving or , from which we deduce that or . These are the two solutions.

A4.

We use the working of A2. to write as

This gives

So taking square roots, or from which we deduce that or . These are the two solutions.

A5. Using the quadratic formula we have:

and

A6. Solving equation we have zeroes

and

hence

You are now in a position to systematically solve all quadratic factorisation problems, without any guess work. Thank you al Khwarizmi!

Share this article:

This article is from the free online course:

Maths for Humans: Linear and Quadratic Relations

UNSW Sydney

Get a taste of this course

Find out what this course is like by previewing some of the course steps before you join:

  • A pair of Witchetty Grubs
    Australian bush tucker
    article

    Adventurer Bear Grylls has claimed that, "Pound for pound, insects contain more protein than beef". Let's investigate!

  • The Cartesian plane and the beauty of graph paper
    The Cartesian plane and the beauty of graph paper
    video

    The Cartesian plane, modelled by a sheet of graph paper, is fundamental for much of modern mathematics. Norman Wildberger explains the basics.

  • Quadratics from Apollonius to Bezier
    Quadratics from Apollonius to Bezier
    video

    An introduction to this week's activities on quadratic relations and functions.

  • Painting of Galileo
    Galileo's ball
    article

    What happens to an object, like a ball, when it falls, or better yet, if you toss it in the air? Galileo was the first to understand the answer.

Contact FutureLearn for Support