Question

middle term splitting x^2-√2x-3/2​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

$\sf \: Split \: the \: middle \: terms \: to \: factorize : {x}^{2} - \sqrt{2} x - \dfrac{3}{2}$

$\large\underline{\bold{Solution-}}$

Basic Concept Used :-

Splitting of middle terms :-

In order to factorize  ax² + bx + c we have to find numbers p and q such that p + q = b and pq = ac.

After finding p and q, we split the middle term in the quadratic as px + qx and get desired factors by grouping the terms.

Now, Consider

$\sf \: {x}^{2} - \sqrt{2} x - \dfrac{3}{2}$

On taking 2 as LCM, we have

$= \sf \: \dfrac{1}{2} (2{x}^{2} - 2 \sqrt{2} x - {3})$

Now, in this quadratic polynomial,

we have

• a = 2 and c = - 3, so ac = - 6

Now,

we have to break - 6 in two factors which when added

$\sf \: gives \: - 2 \sqrt{2} \: and \: when \: multiplied \: gives \: - 6$

Now,

$\sf \: - 6 \: can \: be \: break \: down \: as \: - 3 \times 2 = - 3 \times \sqrt{2} \times \sqrt{2}$

can be further re-grouped as

$\sf \: - 6 = (- 3 \sqrt{2}) \times \sqrt{2}$

So, middle terms splitting is represented as

$= \sf \: \dfrac{1}{2} \bigg( 2{x}^{2} + ( - 3 \sqrt{2} + \sqrt{2} )x - {3} \bigg)$

$= \sf \:\dfrac{1}{2} \bigg( 2{x}^{2} - 3 \sqrt{2} x + \sqrt{2} x - {3} \bigg)$

can be rewritten as

$= \sf \: \dfrac{1}{2} \bigg(\sqrt{2} \times \sqrt{2} {x}^{2} - 3 \sqrt{2} x + \sqrt{2} x - {3} \bigg)$

$= \sf \: \dfrac{1}{2} \bigg(\sqrt{2}x \bigg ( \sqrt{2} x - 3 \bigg) + 1 \bigg( \sqrt{2} x - 3 \bigg) \bigg)$

$= \sf \:\dfrac{1}{2} \bigg(\sqrt{2}x - 3\bigg) \bigg(\sqrt{2}x + 1\bigg)$

$= \sf \:\dfrac{1}{ \sqrt{2} \times \sqrt{2} } \bigg(\sqrt{2}x - 3\bigg) \bigg(\sqrt{2}x + 1\bigg)$

$= \sf \: \bigg( \dfrac{ \sqrt{2}x - 3 }{ \sqrt{2} } \bigg) \bigg(\dfrac{ \sqrt{2}x + 1 }{ \sqrt{2} } \bigg)$

$= \sf \: \bigg( x - \dfrac{3}{ \sqrt{2} } \bigg) \bigg( x + \dfrac{1}{ \sqrt{2} } \bigg)$