Question

Find the roof of quadratic equation √2 x
2 +7x +5√2=0 by factorization method

Answer 1

Question :- Find the root of the quadratic equation by factorization method :

$\rm \: \sqrt{2} {x}^{2} + 7x + 5 \sqrt{2} = 0$

$\color{blue}\large\underline{\sf{Solution-}}$

Given quadratic equation is

$\rm \: \sqrt{2} {x}^{2} + 7x + 5 \sqrt{2} = 0$

On splitting the middle terms, we get

$\rm \: \sqrt{2} {x}^{2} + 2x + 5x + 5 \sqrt{2} = 0$

can be rewritten as

$\rm \: (\sqrt{2} {x}^{2} + \sqrt{2}. \sqrt{2} x )+ (5x + 5 \sqrt{2}) = 0$

$\rm \: \sqrt{2}x(x + \sqrt{2}) + 5(x + \sqrt{2}) = 0$

$\rm \: (x + \sqrt{2})(\sqrt{2}x + 5) = 0$

$\rm \: x + \sqrt{2} = 0 \: \: \: or \: \: \: \sqrt{2}x + 5 = 0$

$\rm \: x = \: - \: \sqrt{2} \: \: \: or \: \: \: \sqrt{2}x = - 5$

$\rm \: x = \: - \: \sqrt{2} \: \: \: or \: \: \: x = - \frac{5}{ \sqrt{2} }$

$\rm \: x = \: - \: \sqrt{2} \: \: \: or \: \: \: x = - \frac{5}{ \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} }$

$\rm \: x = \: - \: \sqrt{2} \: \: \: or \: \: \: x = - \frac{5 \sqrt{2} }{2}$

Hence,

$\color{green}\rm\implies \:\boxed{ \rm{ \: x = \: - \: \sqrt{2} \: \: \: or \: \: \: x = - \frac{5 \sqrt{2} }{2} \: \: }}$

$\rule{190pt}{2pt}$

Basic Concept Used :-

Splitting of middle terms :-

In order to factorize  ax² + bx + c we have to find numbers m and n such that m + n = b and mn = ac.

After finding m and n, we split the middle term i.e bx in the quadratic equation as mx + nx and get the required factors by grouping the terms.

$\color{purple}\rule{190pt}{2pt}$

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac