1. **State the problem:** A boy thinks of a number $x$. He squares it and subtracts the original number from the square, resulting in 42. We need to find the number $x$. 2. **Write the equation:** The problem translates to the equation $$x^2 - x = 42$$ 3. **Rearrange the equation:** Move all terms to one side to set the equation to zero: $$x^2 - x - 42 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-1$, and $c=-42$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-1)^2 - 4 \times 1 \times (-42) = 1 + 168 = 169$$ 6. **Find the roots:** $$x = \frac{-(-1) \pm \sqrt{169}}{2 \times 1} = \frac{1 \pm 13}{2}$$ 7. **Evaluate each root:** - $$x = \frac{1 + 13}{2} = \frac{14}{2} = 7$$ - $$x = \frac{1 - 13}{2} = \frac{-12}{2} = -6$$ 8. **Conclusion:** The number the boy thought of can be either $7$ or $-6$. **Final answer:** $$\boxed{7 \text{ or } -6}$$