Question

Factorise: 4x^2-5x-1 by middle term method.

Answer 1

For a quadratic expression of the form $ax^2+bx+c$, factorization by splitting the middle term is a method in which the middle term is split such that the numbers will be the product of $\frac{c}{a}$ and the sum will be equal to $b$

Given: $4x^2-5x+1$

Here $a=4;\ b=-5;\ c=1$

The middle term is to be split such that the product is $(4\times1=4)$ and the sum is $-5$.

The numbers are $-4,-1$.

So the quadratic expression can be written as:

$\implies 4x^2-4x-1x+1$

Taking the common,

$\implies 4x(x-1)-1(x-1)$

$\implies (4x-1)(x-1)$

NB:$4x^2-5x-1$ cannot be factorized by splitting the middle term. S have solved for $4x^2-5x+1$

Answer 2

Given; 4x^2-5x-1

To Find; Factorise: 4x^2-5x+1 by middle term method

Solution; Factorise: 4x^2-5x+1 by middle term method

This question can be easily solved betting the middle term. In this, the middle term is written. in the form of addition and subtraction of multiples numbers

4x^2-5x+1

4x^2-4x-x+1

4x(x-1)-1(x-1)

(x-1)(4x-1)

Hence the factors are x-1 and 4x-1

Answer 3

We recall the real roots of the given quadratic equation by the factorization method. Split the co-efficient of  x  such that $b=p+q$ and $pq=ab$ of a quadratic equation $ax^2+bx+c=0$

Given: Factorise

Here $a=4,\ b=-5$ and $c=1$

Using Middle term method,

$-5=-4-1\\-5=-5$,

$4\times1=-4\times -1\\4=4$

Hence $p=-4,\ q=-1$

$4x^2-5x+1=4x^2-4x-x+1\\=4x(x-1)-1(x-1)\\=(4x-1)(x-1)$                                                                

Hence, the factor of $4x^2-5x+1$ is $4x-1$ and $x-1$.