Question

How do you factor \[{{x}^{2}}-17x+60\]?

Answer 1

Hint: we can solve this question using factorization method. First we will split the middle term into two numbers in such a way that their sum must be equal to the middle term and their product has to be equal to the product of first and last terms of the equation. Then we have to take common terms and then we solve for x values.

Complete step by step answer:
The given expression we have is
\[{{x}^{2}}-17x+60\]
Now we have to split the term in a way that it satisfies the conditions that their sum must be equal to middle term and product must be equal to first and last term.
Here the product is \[1\times 60=60\]
Now, the number \[60\] can be written as \[20\times 3,6\times 10\] or \[12\times 5\].
So, to get \[-17\] as the middle term we will use the factors \[-12\] and \[-5\].
The expression now can be written as
\[\Rightarrow {{x}^{2}}-12x-5x+60\]
We can now take \[x\] common from first two terms and the expression becomes
\[\Rightarrow x\left( x-12 \right)-5x+60\]
Also, we take $5$common from the rest terms and the expression becomes
\[\Rightarrow x\left( x-12 \right)-5\left( x-12 \right)\]
Again, we can see that the term \[\left( x-12 \right)\] can be taken as common from the entire expression as
\[\Rightarrow \left( x-12 \right)\left( x-5 \right)\]
Therefore, we conclude that the factors of the expression \[{{x}^{2}}-17x+60\] are \[\left( x-12 \right)\] and \[\left( x-5 \right)\] and the factors are \[\left( x-12 \right)\left( x-5 \right)\].

Note:
We can also solve this using quadratic formula. We will find the discriminant and substituting in the quadratic formula. We can find the roots and then we can make them as factors also. We can check the factors by finding the values and substituting them in the equation.