Question

How do you simplify $\dfrac{{{x^2} - 3x + 2}}{{2{x^2} - 2x}}$ ?

Answer 1

Hint:
In the given expression the numerator of the given fraction is a quadratic equation and the denominator of the fraction is a polynomial having degree two in one variable. First we will find the factor of the numerator and then we will also factorize the denominator of the fraction. To find the factor of the numerator of the fraction, first we will multiply the absolute term of the quadratic equation with the coefficient of the ${x^2}$ of the quadratic equation. Now we will find the factors of the middle term of the quadratic equation so that on multiplication of these factors, we will find the absolute term of the quadratic equation. Now to factorize the denominator of the fraction, first we will take common from the given polynomial having degree two in one variable.

Complete step by step solution:
Step: 1 the given fraction is,
$\dfrac{{{x^2} - 3x + 2}}{{2{x^2} - 2x}}$
The numerator of the given fraction is,
${x^2} - 3x + 2$
First we will find the factor of the quadratic equation.
The coefficient of the ${x^2}$ is equal to 1 and the constant term of the equation is 2. So the multiplication of the coefficient of ${x^2}$ and constant term is equal to 2. Now we will split middle term of the equation into two parts so that on multiplication it will give the 2. The factor of the middle term is 2 and 1 .
$
   \Rightarrow {x^2} - 3x + 2 \\
   \Rightarrow {x^2} - 2x - 1x + 2 \\
 $
Take common from the equation.
$
   \Rightarrow {x^2} - 2x - 1x + 2 \\
   \Rightarrow x\left( {x - 2} \right) - 1\left( {x - 2} \right) \\
 $
Take $\left( {x - 2} \right)$ as common from the equation.
$ \Rightarrow \left( {x - 2} \right)\left( {x - 1} \right)$
Step: 2 now find the factor of the denominator $2{x^2} - 2x$ .
Take $2x$ as common from the equation.
$
   \Rightarrow 2{x^2} - 2x \\
   \Rightarrow 2x\left( {x - 1} \right) \\
 $
Therefore the given expression can be written as,
$ \Rightarrow \dfrac{{\left( {x - 2} \right)\left( {x - 1} \right)}}{{2x\left( {x - 1} \right)}}$
Cancel the same term from numerator and denominator of the expression.
$ \Rightarrow \dfrac{{\left( {x - 2} \right)}}{{2x}}$

Therefore the simple of the given expression is equal to $\dfrac{{\left( {x - 2} \right)}}{{2x}}$

Note:
Students are advised to solve the numerator of the given expression first. They should know to solve the quadratic equation. They must cancel the same term from each denominator and numerator of the given fraction.