1.1 Quadratics.
Welcome to this short ‘insights video’ where learners can better understand how to solve quadratic equations using the ‘completing the square’ technique. This technique can solve ‘perfect square’ quadratic equations like this...
...where the coefficient of the squared term is ‘one’. However, equations do not always appear in this form, for example in the following equation the squared term coefficient is two...
but, the equation can be rearranged like this…
...So, leaving the rest of the equation alone, learners can use the ‘completing the square’ technique to solve the ‘x squared plus two x’ part within the brackets first. An important mistake learners often make is to correctly solve this, but then, forget to replace this back into the original quadratic equation to score full marks.
Simple geometry can help students visualise how the completing the square technique works...
...If our ‘general formula’ is to solve x squared plus b times x. Learners can think of this equation as two rectangles and we need to find out the area!
The first rectangle is x wide by x height. It’s area represents the x2 part of the equation, and is a square. The second rectangle is b long and x wide. It’s area represents the b times x part of the equation. Now the learners could rearrange these shapes like this…
And if we were to ‘complete the square’ it would look like this…
So, the total area is easy to work out. It’s x plus b over two all squared. But we don't want the blue square - it’s not part of our equation. So we have to subtract b over two, times b over two. Which is b squared over four. So, the answer to the general solution becomes...
Then learners can replace this general solution back in their original equation within the brackets. And because it only contains one x function now the original quadratic equation is easy to rearrange.
I hope this short insights video has been useful to you to help explain to your learners the types of equations ‘completing the square’ solves and a very visual way to explain how to use the ‘completing the square’ method.
Thank you.
Video transcript1.1 Quadratics.
