Related Formula
The definition of modulus function handles sub-intervals via critical points:
|x - a| = cases x - a & if x ≥ a -(x - a) & if x < a cases$$|x - a| = \begin{cases} x - a & \text{if } x \ge a \\ -(x - a) & \text{if } x < a \end{cases}$$
Core Logic
The critical points are x = 2$x = 2$ and x = 3$x = 3$. We check the three distinct structural intervals:
Case I: x < 2$x < 2$
x(-(x - 2)) + 3(-(x - 3)) + 1 = 0$$x(-(x - 2)) + 3(-(x - 3)) + 1 = 0$$
-x² + 2x - 3x + 9 + 1 = 0 x² + x - 10 = 0$$-x^2 + 2x - 3x + 9 + 1 = 0 \implies x^2 + x - 10 = 0$$
x = -1 ± √(1 + 40)2 = -1 ± √(41)2$$x = \frac{-1 \pm \sqrt{1 + 40}}{2} = \frac{-1 \pm \sqrt{41}}{2}$$
Checking domain constraint x < 2$x < 2$:
-1 - √(41)2 ≈ (-1 - 6.4)/(2) = -3.7 < 2$\frac{-1 - \sqrt{41}}{2} \approx \frac{-1 - 6.4}{2} = -3.7 < 2$ (Valid root)
-1 + √(41)2 ≈ (-1 + 6.4)/(2) = 2.7 < 2$\frac{-1 + \sqrt{41}}{2} \approx \frac{-1 + 6.4}{2} = 2.7 \not< 2$ (Rejected)
Step 1: Intermediate Interval Check
Case II: 2 ≤ x < 3$2 \le x < 3$
x(x - 2) + 3(-(x - 3)) + 1 = 0$$x(x - 2) + 3(-(x - 3)) + 1 = 0$$
x² - 2x - 3x + 9 + 1 = 0 x² - 5x + 10 = 0$$x^2 - 2x - 3x + 9 + 1 = 0 \implies x^2 - 5x + 10 = 0$$
Discriminant check: D = (-5)² - 4(1)(10) = 25 - 40 = -15 < 0$D = (-5)^2 - 4(1)(10) = 25 - 40 = -15 < 0$.
No real roots exist in this interval.
Step 2: Upper Interval Check
Case III: x ≥ 3$x \ge 3$
x(x - 2) + 3(x - 3) + 1 = 0$$x(x - 2) + 3(x - 3) + 1 = 0$$
x² - 2x + 3x - 9 + 1 = 0 x² + x - 8 = 0$$x^2 - 2x + 3x - 9 + 1 = 0 \implies x^2 + x - 8 = 0$$
x = -1 ± √(1 + 32)2 = -1 ± √(33)2$$x = \frac{-1 \pm \sqrt{1 + 32}}{2} = \frac{-1 \pm \sqrt{33}}{2}$$
Checking domain constraint x ≥ 3$x \ge 3$:
-1 + √(33)2 ≈ (-1 + 5.74)/(2) = 2.37 < 3$\frac{-1 + \sqrt{33}}{2} \approx \frac{-1 + 5.74}{2} = 2.37 < 3$ (Rejected)
-1 - √(33)2 < 0$\frac{-1 - \sqrt{33}}{2} < 0$ (Rejected)
Thus, only 1 valid real root satisfies the conditional layout across all ranges.
Pattern Recognition
Always perform case-by-case boundaries checks on algebraic roots found inside absolute modulus problems to discard ghost solutions quickly.
Chapter Mix
Class 11 Mathematics: Quadratic Equations
