waht is the answer to |X+1| = x^2-5

Question

solving equations involving absolute value

Answer 1

|x+1|=x²-5. If x+1>0, that is, x>-1, then:

x+1=x²-5, x²-x-6=(x-3)(x+2)=0, so x-3=0, x=3; or x+2=0, x=-2. Therefore, since x must be greater than -1 we reject x=-2. That leaves x=3 as one solution. Plugging in x=3: |3+1|=4=9-5✔️ This confirms that x=3 is a solution

If x+1<0, x<-1 and -(x+1)=x²-5, -x-1=x²-5, x²+x-4=0, x=(-1±√(1+16))/2=1.5616 or -2.5616. So we reject the positive root because we need x+1 to be negative. That leaves x=-2.5616 as the other solution.

So the solutions are x=3 or -½(1+√17). CHECK: |-2.5616+1|=1.5616; x²-5=1.5616. This result checks out and confirms the second solution.