Sum

Find the roots of the following quadratic equations by factorisation

`sqrt2 x^2 +7x+ 5sqrt2 = 0`

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#### Solution 1

`sqrt2 x^2 + 7x + 5sqrt2`

= `sqrt2 x2 + 5x + 2x + 5sqrt2`

= `x (sqrt2x + 5) + sqrt2(sqrt2x + 5)= (sqrt2x + 5)(x + sqrt2)`

Roots of this equation are the values for which `(sqrt2x + 5)(x + sqrt2) = 0`

`:.sqrt2x + 5 = 0 or x + sqrt2 = 0`

⇒ `x = (-5)/sqrt2 or x = -sqrt2`

#### Solution 2

We write 7x = 5x + 2x as `sqrt2x^2 xx 5sqrt2 = 10x^2 = 5x xx 2x`

`:.sqrt2x^2 + 7x + 5sqrt2 = 0`

`=>sqrt2x^2+ 5x + 2x + 5sqrt2 = 0`

`=> x(sqrt2x + 5) + sqrt2(sqrt2x + 5) = 0`

`=> (sqrt2x + 5)(x + sqrt2) = 0`

`=> x + sqrt2 = 0`or `sqrt2x + 5 = 0`

`=> x = -sqrt2 or x = - 5/sqrt2 = -(5sqrt2)/2`

Hence the roots of the given equation are `-sqrt2 and - (5sqrt2)/2`

Concept: Solutions of Quadratic Equations by Factorization

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