Question

Solve each equation. Identify the solutions as rational or irrational numbers.

Answer 1

`\textbf{a)}\\\\~~~~~~~x(3x-4)=5\\\\\implies 3x^2 -4x = 5\\\\\implies 3x^2 -4x=5\\\\\implies x^2 - \frac 43 x = \frac 53\\ \\\implies x^2 - 2\cdot \frac 23 \cdot x + \left( \frac 23 \right)^2 - \left( \frac 23 \right)^2 = \frac 53\\\\\implies \left(x-\frac 23 \right)^2 = \frac 53 + \frac 49\\\\\implies \left(x-\frac 23 \right)^2= (19)/(9)\\\\\implies x- \frac 23 = \pm\frac{√(19)}3\\\\\implies x=\frac 23 \pm\frac{√(19)}3\\\\`

`\text{The solutions are irrational.}\\\\\\\textbf{b)}\\\\~~~~~~~(3x-2)^2 = \frac 14\\\\\implies 3x -2 = \pm \frac 12\\\\\implies 3x = 2\pm \frac 12\\\\\implies x = \frac 23 \pm \frac 16\\\\ \implies x = \frac 56 ,~~ x =\frac 12 \\ \\\text{The solutions are rational .}`