<PRE><B>Question 1209875</B>
&lt;pre&gt;
Fill in the blanks with constants, to make a true equation:

\frac{x^2 - 6x - 3}{x^3 - 4x} = ___/x + ___/(x - 2) + ___/(x + 2) 

{{{(x^2 - 6x - 3)/(x^3 - 4x)}}} = {{{&quot;___&quot;/x + &quot;___&quot;/(x - 2) + &quot;___&quot;/(x + 2)}}}
{{{(x^2 - 6x - 3)/x(x^2 - 4)}}} = {{{&quot;___&quot;/x + &quot;___&quot;/(x - 2) + &quot;___&quot;/(x + 2)}}}
{{{(x^2 - 6x - 3)/x(x - 2)(x + 2)}}} = {{{&quot;___&quot;/x + &quot;___&quot;/(x - 2) + &quot;___&quot;/(x + 2)}}}
{{{(x^2 - 6x - 3)/x(x - 2)(x + 2)}}} = {{{A/x + B/(x - 2) + C/(x + 2)}}}
{{{matrix(1,3, x^2 - 6x - 3, &quot;=&quot;, A(x - 2)(x + 2) + B(x)(x + 2) + C(x)(x - 2))}}} ---- Multiplying by LCD, x(x - 2)(x + 2)
{{{matrix(1,3, x^2 - 6x - 3, &quot;=&quot;, A(x^2 - 4) + B(x^2 + 2x) + C(x^2 - 2x))}}} 
{{{matrix(1,3, x^2 - 6x - 3, &quot;=&quot;, Ax^2 - 4A + Bx^2 + 2Bx + Cx^2 - 2Cx)}}} 
{{{matrix(1,3, x^2 - 6x - 3, &quot;=&quot;, Ax^2 + Bx^2 + Cx^2 + 2Bx - 2Cx - 4A)}}} 
{{{matrix(1,3, x^2 - 6x - 3, &quot;=&quot;, (A + B + C)x^2 + (2B - 2C)x - 4A)}}}

&lt;font color = blue&gt;&lt;font size = 4&gt;&lt;b&gt;(1)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;&lt;font color = red&gt;&lt;font size = 4&gt;&lt;b&gt;x^2&lt;/font&gt;&lt;/font&gt;&lt;/b&gt; &lt;font color = green&gt;&lt;font size = 4&gt;&lt;b&gt;(- 6)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;&lt;font color = red&gt;&lt;font size = 4&gt;&lt;b&gt;x &lt;/font&gt;&lt;/font&gt;&lt;/b&gt;&lt;font color = magenta&gt;&lt;font size = 4&gt;&lt;b&gt;(- 3)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt; = &lt;font color = blue&gt;&lt;font size = 4&gt;&lt;b&gt;(A + B + C)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;&lt;font color = red&gt;&lt;font size = 4&gt;&lt;b&gt;x^2 &lt;/font&gt;&lt;/font&gt;&lt;/b&gt;&lt;font color = green&gt;&lt;font size = 4&gt;&lt;b&gt;(2B - 2C)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;&lt;font color = red&gt;&lt;font size = 4&gt;&lt;b&gt;x&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;&lt;font color = magenta&gt;&lt;font size = 4&gt;&lt;b&gt; (- 4A)&lt;/font&gt;&lt;/font&gt;&lt;/b&gt;

{{{system(matrix(3,1, matrix(1,6, 1, &quot;=&quot;, A + B + C, &quot;------&quot;, eq, &quot;(i)&quot;), matrix(1,6, - 6, &quot;=&quot;, 2B - 2C, &quot;------&quot;, eq, &quot;(ii)&quot;), matrix(1,6, - 3, &quot;=&quot;, - 4A, &quot;---------&quot;, eq, &quot;(iii)&quot;)))}}} ----- EQUATING left-terms coefficients to right-terms coefficients

  &lt;/b&gt;&lt;font color = red&gt;&lt;font size = 4&gt;&lt;b&gt;- 3 = - 4A&lt;/font&gt;&lt;/font&gt;&lt;/b&gt; ---- eq (iii)
  {{{(- 3)/(- 4) = highlight(3/4 = A)}}}

    &lt;/b&gt;&lt;font color = red&gt;&lt;font size = 4&gt;&lt;b&gt;1 = A + B + C&lt;/font&gt;&lt;/font&gt;&lt;/b&gt; -- eq (i)
    {{{matrix(2,1, 1 = 3/4 + B + C, 1 - 3/4 = B + C)}}} ------ Substituting {{{3/4}}} for A 
    {{{1/4 = B + C}}} ----- eq (iv)

  &lt;/b&gt;&lt;font color = red&gt;&lt;font size = 4&gt;&lt;b&gt;- 6 = 2B - 2C&lt;/font&gt;&lt;/font&gt;&lt;/b&gt; ---- eq (ii)
2(- 3) = 2(B - C)
   - 3 = B - C ----- eq (v)

 {{{- 2&amp;3/4 = 2B}}} ----- Adding eqs (iv) &amp; (iv)
 {{{matrix(2,1, - 11/4 = 2B, - 11 = 8B)}}}
{{{highlight(- 11/8 = B))}}} 

     {{{1/4 = B + C}}} ----- eq (iv)
     {{{1/4 = - 11/8 + C}}} -- Substituting {{{- 11/8}}} for B in eq (iv)
 {{{matrix(3,1, 11/8 + 1/4 = C, 11/8 + 2/8 = C, highlight(13/8 = C))}}}

{{{(x^2 - 6x - 3)/x(x - 2)(x + 2)}}} = {{{&quot;___&quot;/x + &quot;___&quot;/(x - 2) + &quot;___&quot;/(x + 2)}}} now becomes: {{{(x^2 - 6x - 3)/x(x - 2)(x + 2)}}} = {{{highlight(highlight_green(highlight((3/4)/x - (11/8)/(x - 2) + (13/8)/(x + 2))))}}}&lt;/pre&gt;</PRE>