Theory of Intelligence • First-Principles AI & STEM Education

Master Summary Table of All Subparts

Part	Original Equation	Method / Substitution	Formed Quadratic	Valid Solution Set	Extraneous Roots
(i)	$\dfrac{1}{3x} + \dfrac{4x}{6} = 1$	Clear LCD $6x$	$2x^2 - 3x + 1 = 0$	$\left\{\dfrac{1}{2}, \; 1\right\}$	None
(ii)	$\dfrac{x}{x+1} + \dfrac{x+1}{x} = \dfrac{5}{2}$	Reciprocal: $y = \dfrac{x}{x+1}$	$2y^2 - 5y + 2 = 0$	$\{-2, \; 1\}$	None
(iii)	$\dfrac{1}{x+1} + \dfrac{2}{x+2} = \dfrac{7}{x+5}$	Cross-multiplication	$2x^2 + x - 3 = 0$	$\left\{1, \; -\dfrac{3}{2}\right\}$	None
(iv)	$\dfrac{a}{ax-1} + \dfrac{b}{bx-1} = a+b$	Grouping: $(a+b-abx)$	Linear branches	$\left\{\dfrac{a+b}{ab}, \; \dfrac{2}{a+b}\right\}$	None
(v)	$3x^2+15x-2\sqrt{x^2+5x+1}=2$	Radical: $u = \sqrt{x^2+5x+1} \ge 0$	$3u^2 - 2u - 5 = 0$	$\left\{\dfrac{1}{3}, \; -\dfrac{16}{3}\right\}$	$u = -1$ rejected
(vi)	$\sqrt{2x+8} + \sqrt{x+5} = 7$	Double squaring	$x^2 - 288x + 1136 = 0$	$\{4\}$	$x = 284$ (squaring artifact)
(vii)	$\sqrt{3x+4} = 2 + \sqrt{2x-4}$	Double squaring	$x^2 - 24x + 80 = 0$	$\{4, \; 20\}$	None
(viii)	$\sqrt{x+5} - \sqrt{x-3} = 2$	Transposition & squaring	Linear in $(x-3)$	$\{4\}$	None