<div class="notes">
 <h2>How to Complete the Square</h2>
 <p>
   A quadratic expression has the general form:  
   $$ax^2 + bx + c, \quad a \neq 0$$
 </p>

 <p>
   <span class="highlight">Completing the square</span> means rewriting the quadratic in the form  
   $$(x + p)^2 + q$$  
   This form is useful for <span class="highlight">solving equations</span> and for finding the maximum/minimum values of a quadratic function.
 </p>

 <!-- Case 1 -->
 <h3>Case 1: When the coefficient of \(x^2\) is 1</h3>
 <ul class="steps">
   <li><b>Step 1:</b> Identify coefficients \(a\), \(b\), and \(c\).</li>
   <li><b>Step 2:</b> Take half the coefficient of \(x\) and square it.</li>
   <li><b>Step 3:</b> Add and subtract this square inside the expression.</li>
   <li><b>Step 4:</b> Rewrite as a perfect square plus/minus a constant.</li>
 </ul>

 <div class="example">
   <h3>Example 1</h3>
   <p><b>Question:</b> Rewrite \(x^2 + 5x + 3\) in completed square form.</p>

   <p><b>Step 1:</b> \(a = 1,\; b = 5,\; c = 3\).</p>
   <p><b>Step 2:</b> Half of \(b = 5\) is \(\tfrac{5}{2}\). Squaring gives \(\left(\tfrac{5}{2}\right)^2 = \tfrac{25}{4}\).</p>
   <p><b>Step 3:</b> Add and subtract \(\tfrac{25}{4}\):  
     $$x^2 + 5x + 3 = \left(x^2 + 5x + \tfrac{25}{4}\right) - \tfrac{25}{4} + 3$$  
     $$= \left(x + \tfrac{5}{2}\right)^2 - \tfrac{13}{4}$$
   </p>
   <p><b>Final Answer:</b>  
     $$x^2 + 5x + 3 = \left(x + \tfrac{5}{2}\right)^2 - \tfrac{13}{4}$$
   </p>
 </div>

 <!-- Case 2 -->
 <h3>Case 2: When the coefficient of \(x^2\) is not 1</h3>
 <p>
   Factor out the coefficient of \(x^2\) first, then complete the square inside the brackets.
 </p>

 <ul class="steps">
   <li><b>Step 1:</b> Factor out the coefficient of \(x^2\).</li>
   <li><b>Step 2:</b> Take half the coefficient of \(x\) inside the bracket and square it.</li>
   <li><b>Step 3:</b> Adjust by subtracting the extra constant, then simplify.</li>
   <li><b>Step 4:</b> Expand back if required.</li>
 </ul>

 <div class="example">
   <h3>Example 2</h3>
   <p><b>Question:</b> Rewrite \(3x^2 - 9x + 50\) in completed square form.</p>

   <p><b>Step 1:</b> Factor out 3:  
     $$3x^2 - 9x + 50 = 3(x^2 - 3x) + 50$$
   </p>

   <p><b>Step 2:</b> Half of \(-3\) is \(-\tfrac{3}{2}\). Squaring gives \(\tfrac{9}{4}\).  
     Add and subtract inside:  
     $$3\left(x^2 - 3x + \tfrac{9}{4}\right) - 3\left(\tfrac{9}{4}\right) + 50$$  
     $$= 3\left(x - \tfrac{3}{2}\right)^2 + \tfrac{173}{4}$$
   </p>

   <p><b>Final Answer:</b>  
     $$3x^2 - 9x + 50 = 3\left(x - \tfrac{3}{2}\right)^2 + \tfrac{173}{4}$$
   </p>
 </div>

 <h2>Summary</h2>
 <p>
   To complete the square for \(ax^2 + bx + c\):  
   <br>• Factor out \(a\) if \(a \neq 1\).  
   <br>• Add and subtract \(\left(\tfrac{b}{2a}\right)^2\).  
   <br>• Rewrite as \((x + \tfrac{b}{2a})^2 + \text{constant}\).
 </p>
</div>