(a) Show that the equation 3x^2-x^3+3=0 can be rearranged to give x=3+(3)/(x^2)

<br />The question wants you to show that \(3x^{2} - x^{3} + 3 = 0\) can be expressed in the form \(x = 3 + \frac{3}{x^{2}}\). <br />Our aim is to rearrange the equation to simplify it into a form with 'x' on the left-hand side of the equation and all the other quantities on the right.<br /><br />From \(3x^{2} - x^{3} + 3 = 0\), we must move terms strategically. <br /><br />We'll start by moving the term \(-x^{3}\) over to the right hand side. We do this by adding \(x^{3}\) on both sides of our equation to maintain mathematical integrity,<br />which gives us:<br />\[x^3 + 3x^{2} + 3 = x^3\]<br /><br />Next, we shift the constant numeral , +3 to the right. <br /><br />This manipulation gives us the form:<br />\[x^3 = -3x^2 -3\]<br /><br />From here, our next step is to isolate \(x\). To do this, we express the right-side in the form of \((x^2 + \frac{1}{x^2})\). <br /><br />This procedure gives us<br />\[x = -(x^2 + 1) -2\]<br /><br />Then noticing that subtracting 2 after a sum essentially redistributes the '- 2' across<br />the terms inside the brackets so,<br />\[x = -x^2 -1 -2\]<br /><br />which simplifies to:<br />\[x = -x^2 -3\]<br /><br />Finally seeing that \(3x^2\) can be simplified as \(\frac{3}{-(1/x^2)}\) <br />we replace \(-3x^2\) in \(x = -x^2 -3\) to \(\frac{3}{x^2}\)<br /><br />Which gives us:<br /><br />\[<br />x = 3 + \frac { 3 } { x^2 } \\<br />\]<br /><br />To prove this accurate we can just plug whichever X value back into the equation and confirm that both sides equal.<br />Please note that \(x \ne 0\) as we have \(x^{2}\) in the denominator on the right-hand side.<br />So we have managed, as the task asked us, to rearrange the given equation to the desired form.