Factorization

1. Factors of \(10\):
   • \((1,10)\)
   • \((2,5)\)

Since factored forms just contain linear factors of a quadratic equation (\(ax^{2}+bx+c\)), it is fairly simple and contain little variation.

Thus we can express the factored form as:

\begin{equation} (x+a)(x+b) \end{equation}

for \(a\) and \(b\) are the linear factors.

The are also various ways to factorize quadratic equations. These include

• Factorization (not universal) [basic]
• Square root property (god help you)
• Square completion (if applicable)
• Quadratic Formula (universal) [memorization needed]

To factorize via factorization, it can be assumed that \(ax^{2}+bx+c\) where \((x+d)(x+e)\).

Hence it is imperative that \(d \times e=a\) (sans-variable) and that \(d+e=b\) (sans-variable).

This method is the most simplest to execute yet is not universal.

This method is more preferable when dealing with simple binomials (ie. \(x^2+16\)). And is used in conjunction with the earlier factorization method.

Whereas unlike in the factorization method, square root property method only requires finding the square root of the number.

A really strong way of factorizing quadratic equations is to complete the square of the equation. This method is utilizes both the previous 2 methods as described, however, is a little more complicated than that.

Assuming \(ax^{2}+bx+c\), the value of \(c\) must be moved to the other side. Then using \(y\) as the factor, \(y\) takes the former position of \(c\), thus making the equation \(ax^{2}+bx+y=c\). The value of \(y\) then can be determined by dividing \(b\) in half. Further proofs should by conducted if needed.

Finishing, the value of \(c\) must be subtracted by \(y\) and then the whole equation should be squared. The final equation should look like \(x=y \pm \sqrt{(c-y)}\)

See Quadratic Formula

However in layman's terms, find both factors by calculating the Quadratic Formula with both minus and plus.

For example, \(x^{2}-10\) is the equation needed to be factored.

1. Find the factors of \(10\)
   • \((1,10)\)
   • \((2,5)\)