Understanding the Absolute Value Equation

We are asked to find the sum of all real roots of the equation: $|x - 3|^2 + |x - 3| - 2 = 0$.

This equation involves the absolute value function, denoted by $|...|$. Remember that the absolute value of a number is its distance from zero on the number line, and it is always non-negative.

Solving the Equation Using Substitution

To simplify the equation, we can use a substitution. Let $y = |x - 3|$.

Since $y = |x - 3|$, we know that $y \ge 0$ because the absolute value is always non-negative.

Substituting $y$ into the original equation, we get a quadratic equation in terms of $y$:

$y^2 + y - 2 = 0$

Finding Solutions for the Quadratic Equation

We can solve this quadratic equation for $y$ by factoring, using the quadratic formula, or completing the square. Factoring is often the easiest method if applicable.

We look for two numbers that multiply to -2 and add up to +1. These numbers are +2 and -1.

So, the quadratic equation can be factored as:

$(y + 2)(y - 1) = 0$

Setting each factor equal to zero gives the possible values for $y$:

• $y + 2 = 0 \implies y = -2$
• $y - 1 = 0 \implies y = 1$

Determining Valid Values for |x - 3|

We defined $y = |x - 3|$. As we discussed earlier, the absolute value of a real number cannot be negative. Therefore, $|x - 3|$ must be greater than or equal to zero ($|x - 3| \ge 0$).

Let's check the values of $y$ we found:

• $y = -2$: This value is negative. Since $y = |x - 3|$ must be non-negative, $y = -2$ is not a valid solution for $|x - 3|$.
• $y = 1$: This value is positive. This is a valid solution for $|x - 3|$.

So, the only valid condition we need to consider is $|x - 3| = 1$.

Finding the Real Roots (Values of x)

Now we solve the equation $|x - 3| = 1$ for $x$.

The equation $|a| = b$ (where $b \ge 0$) means that $a = b$ or $a = -b$.

Applying this to $|x - 3| = 1$, we have two possibilities:

• Case 1: $x - 3 = 1$
• Case 2: $x - 3 = -1$

Solving Case 1:

$x - 3 = 1$

Add 3 to both sides:

$x = 1 + 3$

$x = 4$

Solving Case 2:

$x - 3 = -1$

Add 3 to both sides:

$x = -1 + 3$

$x = 2$

The real roots of the original equation are $x = 4$ and $x = 2$.

Calculating the Sum of Real Roots

The question asks for the sum of all real roots. The real roots we found are 4 and 2.

Sum of roots = $4 + 2$

Sum of roots = $6$

The sum of all real roots of the equation $|x - 3|^2 + |x - 3| - 2 = 0$ is 6.

Revision Table: Equation Roots

Additional Information: Absolute Value Equations

Understanding the properties of absolute value is crucial for solving equations and inequalities involving it.

• Definition: For any real number $a$, $|a| = a$ if $a \ge 0$, and $|a| = -a$ if $a < 0$.
• Non-negativity: $|a| \ge 0$ for any real number $a$. This is a key property used in this problem.
• Symmetry: $|-a| = |a|$.
• Solving $|E| = k$: If $k > 0$, the solutions are given by $E = k$ or $E = -k$. If $k = 0$, the solution is $E = 0$. If $k < 0$, there are no real solutions.

In our problem, the substitution $y = |x - 3|$ transformed the equation into a standard quadratic form, making it easier to find possible values for the absolute value expression before finding the values of $x$.