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For a alt=0, determine all real roots of...

For a `alt=0,` determine all real roots of the equation `x^2-2a|x-a|-3a^2=0.`

Text Solution

Verified by Experts

The correct Answer is:
`x={a(1-sqrt2),a (sqrt6-1)}`
Here, `a ge 0`
Given, `x^(2)-2a |x-a|-3a^(2)=0`
Case I When `x ge a`
`impliesx^(2)-2a(x-a)-3a^(2)=0`
`impliesx^(2)-2ax -a^(2)=0`
`impliesx=a +-sqrt2a`
`" "[as a(1+sqrt2)ltaand a (1-sqrt2)gta]`
`therefore"Neglecting" x=a(1+sqrt2)as x gea`
`impliesx=a(1-sqrt2)" "...(i)`
Case II When `x lt a impliesx^(2)+2a(x-a)-3a^(2)=0`
`impliesx^(2)+2ax-5a^(2)=0impliesx=-aa+-sqrt6a`
`" "[as a (sqrt6-1)lta (sqrt6-1)gta]`
`therefore"Neglecting"x=a (-1-sqrt6)impliesx=a(sqrt6-1)" "...(ii)`
From Eqs. (i) and (ii), we get
`x={a(1-sqrt2),a (sqrt6-1)}`
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