Question

How do you factor the expression: $2{x^2} - 9x + 4?$

Answer 1

Hint: We will first write the general quadratic polynomial and then, multiply the coefficient of square of x and constant term and then factor the coefficient of x in terms of that and now do the required modifications to get the factors.

Complete step-by-step answer:
We are given that we need to factorize $2{x^2} - 9x + 4$.
The general form of a quadratic polynomial is given by $a{x^2} + bx + c$.
Now, if we compare the given equation to the general equation, we get: a = 2, b = -9 and c = 4.
Now, if we multiply a and c, we have 8.
We can write 8 as 8 times 1.
So, we can write the given polynomial as following:-
\[ \Rightarrow 2{x^2} - 9x + 4 = 2{x^2} - x - 8x + 4\]
Taking x common from first two terms and -4 common from the last two terms, we will then get:-
\[ \Rightarrow 2{x^2} - 9x + 4 = x\left( {2x - 1} \right) - 4\left( {2x - 1} \right)\]
Since, both the terms have 2x – 1 in common, taking it out, we will get:-
\[ \Rightarrow 2{x^2} - 9x + 4 = \left( {2x - 1} \right)\left( {x - 4} \right)\]

Hence, we have factored the required expression.

Note:
The students must note that there is an alternate way to do the same.
Let us assume this to be a quadratic equation: $2{x^2} - 9x + 4 = 0$
Now, if we compare it to the general quadratic equation, which is given by $a{x^2} + bx + c = 0$ whose roots are given by:- $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, we get a = 2, b = -9 and c = 4.
Let us put these values in the formula mentioned above so that we get the following roots:
\[ \Rightarrow x = \dfrac{{ - ( - 9) \pm \sqrt {{{( - 9)}^2} - 4 \times 2 \times 4} }}{{2 \times 2}}\]
Simplifying the calculations a bit to get:-
\[ \Rightarrow x = \dfrac{{9 \pm \sqrt {81 - 32} }}{4}\]
Simplifying the calculations a bit further to get:-
\[ \Rightarrow x = \dfrac{{9 \pm \sqrt {49} }}{4}\]
We know that square of 7 is 49.
\[ \Rightarrow x = \dfrac{{9 \pm 7}}{4}\]
Hence, the possible roots are 4 and $\dfrac{1}{2}$.
Now, since the roots are found, our equation will be $(x - 4)\left( {x - \dfrac{1}{2}} \right) = 0$.
We can write this equation as $(x - 4)\left( {\dfrac{{2x - 1}}{2}} \right) = 0$
Multiplying the whole equation by 2, we get the required answer.
Therefore, the equation will be (2x – 1)(x – 4).