Question

The sum of all the roots of the equation \((x-1)^2 - 5|x-1| + 6 = 0\), is:

• A. 5
• B. 3
• C. 4
• D. 1

Hint:

For equations involving \(|f(x)|\) and \(f(x)^2\), the substitution \(y=|f(x)|\) is a very effective technique. It transforms the equation into a standard polynomial, which is easier to solve. Always remember to substitute back and solve for the original variable, considering both positive and negative cases for the absolute value.

Answer 1

The Correct Option is C

To solve the equation \((x-1)^2 - 5|x-1| + 6 = 0\), we need to consider two cases based on the nature of the absolute value function, which depends on whether the expression inside the absolute value is positive or negative. 

1. Case 1: \(x-1 \geq 0\), i.e., \(x \geq 1\)
   • When \(x-1 \geq 0\), we have \(|x-1| = x-1\).
   • The equation becomes:
   • Let \(y = x - 1\). Then, the equation simplifies to:
   • Solving this quadratic equation by factoring:
   • The solutions are \(y = 2\) and \(y = 3\).
   • Since \(y = x - 1\), substituting back gives \(x = 3\) and \(x = 4\).
2. Case 2: \(x-1 < 0\), i.e., \(x < 1\)
   • When \(x-1 < 0\), we have \(|x-1| = -(x-1)\).
   • The equation becomes:
   • Let \(z = x - 1\). Then, the equation simplifies to:
   • Solving this quadratic equation by factoring:
   • The solutions are \(z = -2\) and \(z = -3\).
   • Since \(z = x - 1\), substituting back gives \(x = -1\) and \(x = -2\).
   • However, these values are not valid for \(x < 1\) as they satisfy \(x - 1 < 0\), but the inequality condition should be checked. For \(x = -1\) and \(x = -2\), \(x - 1\) indeed satisfies the condition being less than zero; hence they verify as valid solutions in the secondary check of inequality after the substitution.

Now, combining the valid solutions from both cases, we have \(x = 3, 4, -1, -2\).

To find the sum of all these roots, calculate:

\((3 + 4 + (-1) + (-2)) = 4\)

Thus, the sum of all the roots is 4.