Question

Factorise the following:-

Answer 1

${a}^{2} + a + \frac{1}{4}$

This question isnot in standard form of polynomial.

STANDARD FORM OF POLYNIMIAL = ax^3 + bx^2 + cx^1 + dx^0 .....

so, we can convert the fraction 1/4 by multiplying the polynomial by 4.

$4 ({a}^{2} + a + \frac{1}{4} ) \\ \\ = 4 {a}^{2} + 4a + 1$

Now we can easily factorise it by middle term factorisation.

The middle term should be splitted in such a way that their sum gives the middle term while their multiplication gives the value equal to product of coefficient of x^2 and x^0.

$4 {a}^{2} + 4a + 1 \\ \\ = 4 {a}^{2} + 2a + 2a + 1 \\ \\ = 2a(2a + 1) + 1(2a + 1) \\ \\ = (2a + 1)(2a + 1) = {(2a + 1)}^{2}$

VERIFICATION :-

${(2a + 1)}^{2} = 4 {a}^{2} + 2a + 1$