2.1. Given \( \frac{4 x-3}{x^{2}}=m \), if -4 is one of the roots of the equation, determine the value of \( m \) and the other root.

Solve the equation \( 4x-3=m*x^{2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(4x-3=mx^{2}\) - step1: Move the expression to the left side: \(4x-3-mx^{2}=0\) - step2: Rewrite in standard form: \(-mx^{2}+4x-3=0\) - step3: Solve using the quadratic formula: \(x=\frac{-4\pm \sqrt{4^{2}-4\left(-m\right)\left(-3\right)}}{2\left(-m\right)}\) - step4: Simplify the expression: \(x=\frac{-4\pm \sqrt{4^{2}-4\left(-m\right)\left(-3\right)}}{-2m}\) - step5: Simplify the expression: \(x=\frac{-4\pm \sqrt{16-12m}}{-2m}\) - step6: Simplify the expression: \(x=\frac{-4\pm 2\sqrt{4-3m}}{-2m}\) - step7: Separate into possible cases: \(\begin{align}&x=\frac{-4+2\sqrt{4-3m}}{-2m}\\&x=\frac{-4-2\sqrt{4-3m}}{-2m}\end{align}\) - step8: Simplify the expression: \(\begin{align}&x=\frac{2-\sqrt{4-3m}}{m}\\&x=\frac{-4-2\sqrt{4-3m}}{-2m}\end{align}\) - step9: Simplify the expression: \(\begin{align}&x=\frac{2-\sqrt{4-3m}}{m}\\&x=\frac{2+\sqrt{4-3m}}{m}\end{align}\) The roots of the equation \(4x-3=m*x^{2}\) are given by the solutions to the quadratic equation \(4x-3=m*x^{2}\). Given that -4 is one of the roots, we can substitute \(x=-4\) into the equation to find the value of \(m\). Substituting \(x=-4\) into the equation \(4x-3=m*x^{2}\), we get: \[4(-4)-3=m*(-4)^{2}\] Simplifying this equation, we have: \[-16-3=16m\] \[-19=16m\] \[m=-\frac{19}{16}\] Therefore, the value of \(m\) is \(-\frac{19}{16}\). To find the other root, we can substitute \(m=-\frac{19}{16}\) back into the solutions for \(x\): \[x=\frac{2-\sqrt{4-3m}}{m}\] \[x=\frac{2-\sqrt{4-3(-\frac{19}{16})}}{-\frac{19}{16}}\] \[x=\frac{2-\sqrt{4+\frac{57}{16}}}{-\frac{19}{16}}\] \[x=\frac{2-\sqrt{\frac{64+57}{16}}}{-\frac{19}{16}}\] \[x=\frac{2-\sqrt{\frac{121}{16}}}{-\frac{19}{16}}\] \[x=\frac{2-\frac{11}{4}}{-\frac{19}{16}}\] \[x=\frac{\frac{8-11}{4}}{-\frac{19}{16}}\] \[x=\frac{-\frac{3}{4}}{-\frac{19}{16}}\] \[x=\frac{3}{4} \times \frac{16}{19}\] \[x=\frac{12}{19}\] Therefore, the other root of the equation is \(\frac{12}{19}\).