Question

3x^2+2root5x-5=0
Completing the square method
I am supposed to get answer as root 5/3

Answer 1

x = ±
`(√(5) )/(3)`

Explanation:

Given

3x² + 2
`√(5)` x - 5 = 0 ( add 5 to both sides )

3x² + 2
`√(5)` x = 5

Before completing the square we require the coefficient of the x² term to be 1

Factor out 3 from each term on the left side

3(x² +
`(2√(5) )/(3)` x ) = 5

To complete the square

add/ subtract ( half the coefficient of the x- term)² to x² +
`(2√(5) )/(3)` x

3(x² + 2(
`(√(5) )/(3)` )x +
`(5)/(9)` -
`(5)/(9)` ) = 5

3(x +
`(√(5) )/(3)` )² + ( 3 × -
`(5)/(9)` ) = 5

3(x +
`(√(5) )/(3)` )² -
`(5)/(3)` = 5 ( add
`(5)/(3)` to both sides )

3(x +
`(√(5) )/(3)` )² =
`(20)/(3)` ( divide both sides by 3 )

(x +
`(√(5) )/(3)` )² =
`(20)/(9)` ( take the square root of both sides )

x +
`(√(5) )/(3)` = ±
`\sqrt{(20)/(9) }` = ±
`(2√(5) )/(3)` ( subtract
`(√(5) )/(3)` from both sides )

x = -
`(√(5) )/(3)` ±
`(2√(5) )/(3)`

Thus

x = -
`(√(5) )/(3)` -
`(2√(5) )/(3)` = -
`(√(5) )/(3)`

x = -
`(√(5) )/(3)` +
`(2√(5) )/(3)` =
`(√(5) )/(3)`