absolute value equations

Question

Solve the equations:
|x|= x^2+x-24
|x+1|=x^2-5

Answer 1

There are two possible values, located at the points where the functions intersect.

 

|x|= x^2+x-24

 

x=x^2+x-24

0=x^2-24

0=(x+2*√(6))*(x-2*√(6))

 

-x=x^2+x-24

0=x^2+2*x-24

0=(x+6)*(x-4)

 

So for the first equation, the solutions are 

x = 2*√(6) or x = -6

 

For the second equation

 

x+1=x^2-5

0=x^2-x-6

0=(x-3)*(x+2)

 

-(x+1)=x^2-5

0=x^2+x-4

0=(x+1/2*√(17)+1/2)*(x-1/2*√(17)+1/2)

 

x = 3 or x = -1/2*√(17)+1/2