Question

How do you simplify \[\dfrac{{{x^3} - 9x}}{{{x^2} - 7x + 12}}\]?

Answer 1

Hint: In the above question, is based on simplifying the algebraic expression. The main approach towards this question is to apply the formula of factors of difference of sum in the numerator. For that every term in the numerator should be a perfect square and then further the common terms get cancelled in the numerator and denominator.

Complete step by step answer:
Algebraic expression in mathematics is an expression which is made up of variables, constants along with mathematical operations like addition, subtraction, division and multiplication. So, the combination of terms is said to be an expression.We need to simplify the above expression. It can be done by reducing the highest power in the numerator i.e, \[x\left( {{x^2} - 9} \right)\].

We have taken the variable x common in the numerator which will reduce the power inside the bracket from 3 to 2.Now we get the difference of square and factors are further written in the form,
\[{a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)\]
Now writing it in the above form,
\[{\left( x \right)^2} - {\left( 3 \right)^2}\]
where a=x and b=3
\[{\left( x \right)^2} - {\left( 3 \right)^2} = \left( {x - 3} \right)\left( {x + 3} \right)\]
Then writing the whole numerator,
\[x\left( {{x^2} - 9} \right) = x\left( {x - 3} \right)\left( {x + 3} \right)\]
Now by factoring the expression in denominator by equating it with zero we get,
\[{x^2} - 7x + 12 = 0\]
Factors of \[12{x^2}\] should be calculated in such a way that their addition should be equal to -7x.
\[
{x^2} - 7x + 12 = \left( {x - 3} \right)\left( {x - 4} \right) \\
\Rightarrow \dfrac{{{x^3} - 9x}}{{{x^2} - 7x + 12}} = \dfrac{{x\left( {x - 3} \right)\left( {x + 3} \right)}}{{\left( {x - 3} \right)\left( {x - 4} \right)}} \\ \]
Now on cancelling the common factor we obtain,
\[\dfrac{{x\left( {x + 3} \right)}}{{\left( {x + 4} \right)}}\]

Note:The quadratic equation in the denominator, an alternative way of finding the factors is by directly solving the equation by using a direct formula which is given below.
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
By substituting the values of a=1, b=-7 and c=12 we get the factors of x.
\[x = \dfrac{{ - \left( { - 7} \right) \pm \sqrt {{{\left( { - 7} \right)}^2} - 4\left( 1 \right)\left( {12} \right)} }}{{2\left( 1 \right)}}\]. Therefore, the value we get of x is -4.