Question

How do you factorise the given equation $2{x^2} - 5x + 1$?

Answer 1

Hint: In this question we need to find the factor given algebraic expression. To find the factor of a given algebraic expression we will convert it to a quadratic equation and then find the root of the corresponding quadratic equation. Finally write the factor of the given expression from the root of the quadratic equation.

Complete step by step answer:
Let us try to solve this question in which we are asked to factor the given algebraic expression $2{x^2} - 5x + 1$. To find the factor of this quadratic, we will first find the discriminant of this quadratic equation and from which we find the nature of the root of this quadratic equation. After which we will find the root of the quadratic equation using quadratic formula and factor it.

Types of root of quadratic equation: $a{x^2} + bx + c = 0$
1. Two distinct real roots, if ${b^2} - 4ac > 0$ (which is called discriminant of this quadratic equation)
2. Two equal real roots, if ${b^2} - 4ac = 0$
3. No real roots if, ${b^2} - 4ac < 0$
In the given quadratic equation $2{x^2} - 5x + 1 = 0$, we have
$
\Rightarrow a = 2 \\
\Rightarrow b = - 5 \\
\Rightarrow c = 1 \\
 $
Discriminant of the quadratic equation is
$ \Rightarrow {b^2} - 4ac = \,{( - 5)^2} - 4 \cdot 2 \cdot 1 \\
\Rightarrow 25 - 8 \\
\Rightarrow 17 > 0
 $
Hence the given quadratic equation has two distinct real roots.

Now, root of this equation by using quadratic formula are,
$
\Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}
\Rightarrow \,\dfrac{{ - ( - 5) \pm \sqrt {{{( - 5)}^2} - 4 \cdot 2 \cdot 1} }}{{2 \cdot 2}} \\
\Rightarrow dfrac{{5 \pm \sqrt {25 - 8} }}{4} \\
\Rightarrow dfrac{{5 \pm \sqrt {17} }}{4} \\
 $
Hence the root of quadratic equation $2{x^2} - 5x + 1 = 0$ are $x = \dfrac{{5 + \sqrt {17} }}{4}$ and $x = \dfrac{{5 - \sqrt {17} }}{4}$.

Hence the given expression $2{x^2} - 5x + 1$ factor will be $\left( {x - \left( {\dfrac{{5 + \sqrt {17} }}{4}} \right)} \right)$ and $\left( {x - \left( {\dfrac{{5 - \sqrt {17} }}{4}} \right)} \right)$.

Note: While solving these types of questions we will first change the given expression into a quadratic equation and then check the nature of the root and then find the root of that quadratic equation using quadratic formula. To solve this type of question you need to know the conditions for the nature of roots.