Factorize: \[{x^2} - 8x + 15\]

Hint: In order to solve this question we have to find the value of \[x\] by the factorization method:
Here we have to split the coefficient of \[x\] in such a way that the product of those terms is the product of the coefficient of \[{x^2}\] and the constant term. Then take commonly from those terms and equate those to zero in order to get the value of \[x\].

Complete step-by-step answer:
Given,
The quadratic equation, \[{x^2} - 8x + 15\]
To find,
The roots of the \[{x^2} - 8x + 15\]
The given equation is \[{x^2} - 8x + 15 = 0\]
Now we have to split the coefficient of \[x\] including \[x\] such that the product of that term is the multiplication of coefficient of \[{x^2}\] and constant including \[{x^2}\].
So, \[ - 8x\] is split in such a way that their product is \[15{x^2}\] and addition is \[ - 8x\]
On splitting \[ - 8x\] the new terms are \[ - 3x\] and \[ - 5x\]
On applying this condition to the equation
\[{x^2} - 8x + 15 = 0\]
On splitting the middle term
\[{x^2} - 3x - 5x + 15 = 0\]
Now taking something common
\[x\left( {x - 3} \right) - 5\left( {x - 3} \right) = 0\]
Now taking \[x - 3\] common
\[\left( {x - 5} \right)\left( {x - 3} \right) = 0\]
On equating both these equations \[ = 0\] then we are able to find the roots of that equation.
On equating to zero. The roots of the equation are
\[x = 5\] and
\[x = 3\]
The roots of the equation: \[{x^2} - 8x + 15\]
\[ \Rightarrow x = 5\] and
\[ \Rightarrow x = 3\]

Note: To solve these type of question first we have to split the term containing in such a way that the sum of that term is the term that contains \[x\] and the product of that term is the product of term that contains \[{x^2}\] and the constant term. And then take commonly from that and make two terms of that. And again take the common term from those two terms and equate both those terms to zero in order to get the roots of the equation. The degree of polynomial decides the number of roots of the equation.