Solving Quadratic Equations

1. By factorisation:

   Example: Solve \(x^2 - 5x + 6 = 0\).

   \(x^2 - 5x + 6 = 0\)

   \((x - 2)(x - 3) = 0\)

   Since the product is zero, at least one factor must be zero.

   \(x - 2 = 0 \quad \text{or} \quad x - 3 = 0\)

   \(x = 2 \quad \text{or} \quad x = 3\)

2. By completing the squares:

   When the quadratic equation is in the general form \(ax^2 + bx + c = 0\)

   Step I: Factorise the coefficient of \(x^2\)

   \( a \left[ x^2 + \frac{b}{a}x \right] + c = 0\)

   Step II: Complete the square - Add and Subtract \( \left( \frac{b}{2a} \right)^2 \)

   \(a \left[ x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 \right] + c = 0\)

   Step III: Write the quadratic as a perfect square

   \(a \left[ \left(x + \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 \right] + c = 0 \)

   \(a \left(x + \frac{b}{2a} \right)^2 - a \left( \frac{b}{2a} \right)^2 + c = 0 \)

   \(a \left(x + \frac{b}{2a} \right)^2 - \frac{b^2}{4a} + c = 0 \)

   Step IV: Make the perfect square the subject

   \(\left(x + \frac{b}{2a} \right)^2 = \frac{b^2}{4a^2} - \frac{c}{a}\)

   Step V: Solve for \(x\)

   \(x = \pm\sqrt{\frac{b^2}{4a^2} - \frac{c}{a}} - \frac{b}{2a} \)

   \(x = \pm\sqrt{ \frac{b^2 - 4ac}{4a^2} } - \frac{b}{2a} \)

   \(x = \frac{\pm \sqrt{b^2 - 4ac} }{2a} - \frac{b}{2a} \)

3. By quadratic formula:

   The quadratic equation must be expressed in the general form of \(ax^2 + bx + c = 0\), where \(a \neq 0\)

   using the quadratic formula,

   \(x = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a} \)

In the completing-square derivation, \(a\ne0\). For real roots, require \(b^2-4ac\ge0\).