91Ó°ÊÓ

Absolute value equations involve expressions within the absolute value operator, which represents the distance from zero on a number line. In simple terms, absolute value always results in a non-negative number. For example, \(|x-1|\) means you find the distance from \(x\) to 1.

When solving absolute value equations like \(a = |x|\), two scenarios arise:

• Set the expression inside the absolute value equal to \(a\): \(x = a\).
• Set the expression inside the absolute value equal to the negative of \(a\): \(x = -a\).

Consider each case separately to find all possible solutions. By doing this step in the problem \( (x-1)^{2} - 5|x-1| + 6 = 0 \), we got two scenarios based on the value of \(x-1\).

This split approach is essential when dealing with absolute values as it helps account for both positive and negative possibilities.

Quadratic equations are mathematical expressions of the form \( ax^{2} + bx + c = 0 \). The solutions to these equations are called roots, and they represent the values of \(x\) that satisfy the equation.

To solve a quadratic equation, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The term under the square root, \(b^2 - 4ac\), is called the discriminant.

• If the discriminant is positive, the equation has two distinct real roots.
• If it's zero, there's exactly one real root.
• If it's negative, the roots are complex numbers.

In the exercise, both cases for solving \(x^2 - 7x + 12 = 0\) resulted in finding real roots using basic factoring. The equation was factored to find \(x = 3\) and \(x = 4\), both of which are real and meet the problem's conditions.

The number of solutions refers to how many distinct values satisfy a given equation. For absolute value and quadratic equations, assessing conditions for solutions is crucial.

In the exercise, once the absolute value expression \(|x-1|\) was dealt with, determining the number of valid solutions involved checking each case separately. Only solutions that fit the constraints (\(x-1 \geq 0\) or \(x-1 < 0\)) were accepted.

For \(x^{2} - 7x + 12 = 0\), \(x=3\) and \(x=4\) were solutions. Both values adhere to the condition \(x-1\geq0\), so these were counted. There were no valid solutions for \(x-1 < 0\) since the roots did not satisfy the condition.

Thus, the equation \( (x-1)^{2} - 5|x-1| + 6 = 0 \) has two valid roots, \(x=3\) and \(x=4\). Understanding these checks ensures that all potential solutions are accurately identified and counted.