For `a le 0,` jdetermine all real roots of the equation
`x^(2)-2a|x-a|-3a^(2)=0`

Explanation

We have `x^(2)-2a|x-a|-3a^(2)=0`
`because(x^(2)-3a^(2))/(2a)=|x-a|`
Roots of the equation are values of x where the graphs of `because(x^(2)-3a^(2))/(2a)and y=|x-|a` intersect.

From the graph, we have `(x^(2)-3a^(2))/(2a)=a-x" for "x lta`
and `(x^(2)-3a^(2))/(2a)=x-a" for "x gta`
Form (i), `x^(2)-3a^(2)=2a^(2)-2ax`
`impliesx^(2)+2ax-5a=0`
`impliesx=(-2apmsqrt4a^(2)+20a^(2))/(2)`
`=(-2apm2asqrt6)/(2)`
`=-a+asqrt6(asxlta)`
From (ii),
`x^(2)-2a(x-a)3a^(2)=0`
`impliesx^(2)-2ax-a^(2)=0`
`impliesx=(2apmsqrt4a^(2)+4a^(2))/(2)=a-asqrt2(asxgta)`
Thus, the solution set is `(-a+asqrt6,a-asqrt2)`.