Animated Solution for Mathematics - Quadratic Equations: The sum of all the real roots of the equation $|x-2{|}^2+|x-2|-2=0$ is .........

The given equation is $|x-2{|}^2+|x-2|-2=0$. At first glance, this might look like a standard equation, but it is actually a masterclass in disguise.

The secret lies in recognizing the repeating structure. Notice how the term $|x-2|$ appears twice? This is a classic JEE trap designed to test your ability to see patterns.

Instead of panicking, we treat the entire block $|x-2|$ as a single variable. This is the essence of the substitution strategy.

Let us define a new variable, $t=|x-2|$. By making this substitution, our equation transforms into a much friendlier form:

Before you rush to solve this quadratic, there is a vital step that separates the top-tier students from the rest. We must define the domain of our new variable.

Since $t$ is equal to the absolute value of a real number, $t$ must be non-negative. We write this down as the constraint $t\ge 0$. This is your safety net; it will catch any extraneous solutions that might arise later.

Now, we are left with a simple quadratic equation: $t^2+t-2=0$. We need to find two numbers that multiply to $-2$ and add up to $1$.

These numbers are $2$ and $-1$. So, we factor the equation as:

This gives us two potential values for $t$: $t=-2$ or $t=1$. Now, we return to our constraint. We established that $t\ge 0$. Clearly, $t=-2$ violates this condition, so we must reject it. We are left with only one valid solution: $t=1$.

Now that we have $t=1$, we substitute it back into our original definition: $|x-2|=1$. This is where the beauty of geometry comes in.

The equation $|x-2|=1$ tells us that the distance between $x$ and $2$ on the number line is exactly $1$. This means $x$ can be $1$ unit to the right of $2$ (which is $3$) or $1$ unit to the left of $2$ (which is $1$).

Algebraically, we solve this by considering two cases: 1. $x-2=1\Rightarrow x=3$ 2. $x-2=-1\Rightarrow x=1$

The question asks for the sum of all real roots. We have found our roots: $x_1=1$ and $x_2=3$.

Adding them together, we get:

And there you have it! By recognizing the pattern, applying a smart substitution, respecting the constraints, and using geometric intuition, we have conquered this problem. The final answer is 4.