418 /www/nrich/html/content/02/06/15plus3/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><br></br> <p>Show that $x = 1$ is a solution of the equation</p> <p>$$x^{3/2} - 8x^{-3/2} = 7$$</p> <p>Find all the solutions of the equation.</p> <br></br></mdoxml> <mdoxml version="1.0"><br></br> <p>This solution is from Arun Iyer, SIA High School and Junior College.</p> <p>The given equation is $x^{3/2} - 8x^{-3/2} = 7$.</p> <p>Multiply throughout by $x^{3/2}$and rearranging we get</p> <p>$(x^{3/2})^2 - 7x^{3/2} - 8 = 0$</p> <p>This can be factorised as:</p> <p>$(x^{3/2} - 8)(x^{3/2} + 1) = 0$</p> <p>Case 1:</p> <p>Consider,</p> <p>$x^{3/2} + 1 = 0$ or $x^{3/2} = -1$</p> <p>Squaring both sides, $x^3 = 1$ so the three cube roots of unity are solutions.</p> <p>Now</p> <p>$ 1 = e^{2k\pi i} $</p> (for $k=0,1,2,3$...) therefore $ x = e^{{2k\pi i} /3} $ . <p>Putting $k=0$, $1$, $2$ we get:</p> <p>$x = e^0$ (which is $1$) or</p> <p>$ x = exp({\pm({2\pi i} /3)}) $</p> . <p>Case 2:</p> <p>Consider,</p> <p>$x^{3/2}- 8 = 0$ or $x^{3/2} = 8$</p> <p>Squaring both sides, $x^3 = 64$ so the three cube roots of $64$ are solutions.</p> <p>Now</p> <p>$ x^3 = 64 e^{2k\pi i} $</p> <div>(for $k=0,1,2,3$...) therefore</div> <br></br> <div>$ x = 64^{1/3} e^{{2k\pi i}/3} $ .</div> <p>Putting $k=0,1,2$ we get: $x = 4$ or</p> <p>$ x = 4exp({\pm{2\pi i} /3}) $</p> . <p>So the six roots are:</p> <ul> <li>$x_1 = 1$</li> <li>$x_2 = \exp({2\pi i} /3) = \cos (2\pi /3) + i\sin(2\pi /3) = -1/2 + i {\sqrt3}/2 $</li> <li>$x_3 = \exp({-2\pi i} /3) = \cos (-2\pi /3) + i\sin(-2\pi /3) = -1/2 - i {\sqrt3}/2 $</li> <li>$x_4 = 4$</li> <li>$x_5 = 4\exp({2\pi i} /3) = 4\cos (2\pi /3) + 4i\sin(2\pi /3) = -2 + 2i {\sqrt3} $</li> <li>$x_6 = 4\exp({-2\pi i} /3) = 4\cos (-2\pi /3) + 4i\sin(-2\pi /3) = -2 - 2i {\sqrt3} $</li> </ul> <p>Substituting $x=1$ into the given equation we need to recognise that $1^{3/2}$ has two values $+1$ and $-1$ so that whereas $x^{3/2} = 1$ does not satisfy the equation the other value $x^{3/2} = -1$ does satisfy it and hence $x=1$ is a solution of the equation.</p> <br></br></mdoxml> <mdoxml version="1.0"> <p>There are 2 real and 4 complex solutions.</p> </mdoxml> 2 5 0 0 0 0 1 Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and explain why this is so. Find all the solutions of the equation. Algebra Quadratic equations Advanced Algebra Roots of complex numbers Advanced Algebra Complex numbers Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof