Problem Solving

What is the sum of all roots of the equation
|x + 4|^2 - 10|x + 4| = 24?

Please help me find my mistake:
Let x+4=y
Now we get two cases,
Case1:
y^2-10y-24=0
Solving we get -2,12

Case2:
-y^2+10y-24=0
where we get 6,4

uptowngirl92 wrote:What is the sum of all roots of the equation
|x + 4|^2 - 10|x + 4| = 24?

Please help me find my mistake:
Let x+4=y
Now we get two cases,
Case1:
y^2-10y-24=0
Solving we get -2,12

it should be -12,2

Case2:
-y^2+10y-24=0
where we get 6,4

i think this eqn should be y^2+10y-24=0
and roots =12,-2

and now putting back the values in the eqn
x+4=y
when y=-12;x=-16
y=2;x=-2
y=12;x=8
y=-2;x=-6


adding summ of the roots is -16
wats the OA

IMO 20. As in first case, I don't why thats incorrect.

IMO it should be - 8 .. Btw whats the OA?

uptowngirl92 wrote:What is the sum of all roots of the equation
|x + 4|^2 - 10|x + 4| = 24?

Please help me find my mistake:
Let x+4=y
Now we get two cases,
Case1:
y^2-10y-24=0
Solving we get -2,12

Case2:
-y^2+10y-24=0
where we get 6,4

Mind the Co-efficient of the square term in a quadratic equation.

Example:
ax^2 -bx + c = 0

Here 'a' is the co-efficient of the square term.

Sum of roots is always b/a ( its not b)
Product of roots is always c/a ( its not c )

So if you multiple your case 1 with -1 you get case 2. But in your case 2 , signs of a,b and c are opposite of the same in case 1 and they cancel out.
So they are the SAME cases.

In the above light, you are solving the case 2 incorrectly because you ignored the fact that a = -1 and not 1.

You solved the case 1 correclty.