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Question
Find the roots of the following quadratic equation by factorisation:
`sqrt2 x^2 +7x+ 5sqrt2 = 0`
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Solution 1
`sqrt2x^2 + 7x + 5sqrt2`
= `sqrt2 x^2 + 5x + 2x + 5sqrt2`
= `x (sqrt2x + 5) + sqrt2(sqrt2x + 5)`
= `(sqrt2x + 5)(x + sqrt2) = 0`
Roots of this equation are the values for which `(sqrt2x + 5)(x + sqrt2) = 0`
`:.sqrt2x + 5 = 0 orx + 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`
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