The difference of two squares formula provides us with a formula
\[a^2-b^2 = \begin{pmatrix}a+b\end{pmatrix} \begin{pmatrix}a-b\end{pmatrix} \] \[\begin{pmatrix}a+b\end{pmatrix} \begin{pmatrix}a-b\end{pmatrix} = a^2 - b^2\]
Using the difference of two squares formula, distribute each of the following paretheses:
The difference of two squares formula can be used to do quick calculations.
For instance, say we have to calculate \(98\times 102\), without a calculator.
Noticing that each of the two numbers in this product are \(2\) units away from \(100\), we can write:
\[98\times 102 = \begin{pmatrix} 100 - 2 \end{pmatrix} \begin{pmatrix} 100 + 2 \end{pmatrix}\]
We can then use the difference of two squares formula to write:
\[\begin{aligned} 98\times 102 &= \begin{pmatrix} 100 - 2 \end{pmatrix} \begin{pmatrix} 100 + 2 \end{pmatrix} \\
&= 100^2 - 2^2 \\
&= 10000 - 4 \\
& = 9996 \end{aligned}\]
This technique for quick calculations using the difference of two squares is illustrated in the following tutorial.
Use the difference of two squares formula to calculate each of the following without a calculator:
The difference of two squares is often used to factorize expressions.
In particular quadratic equations looking like:
\[x^2-k^2 = 0\]
the trick is to see that, using the difference of two squares formula, this can we written
\[\begin{pmatrix}x-k\end{pmatrix}\begin{pmatrix} x + k \end{pmatrix}\]
Using the difference of two squares formula, write each of the following in factored form:
Using the method described above, we factorize as follows: