Quadratic formula

• If the quadratic \(ax^2 + bx + c = 0\) cannot be factorised (or if you cannot readily determine its factors), the solutions are given by the formula:
  \[ x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\] 
  This formula may be used to obtain the solutions of any quadratic equation.
• For example: solve \(3x^2 - 15x + 17 = 9\) 
  
  Rearranging to give \( 3 x^{2} - 15 x + 8 = 0 \)
  Using the Quadratic formula:
   \begin{eqnarray*}
  x &=& \frac{-b\pm\sqrt{b^2-4ac}}{2a} \\
  &=& \frac{-(-15)\pm\sqrt{(-15)^2-4\times 3 \times 8}}{2\times 3} \\
  &=& \frac{15 \pm \sqrt{129}}{6} \\
  &\approx& 4.39 \mbox{ or } 0.61
  \end{eqnarray*}
  
  
• A quadratic can have:
  • 2 real solutions (roots). This happens when (\(b^2- 4ac\)) is positive.
  • 1 real solution (root). This happens when (\(b^2- 4ac\)) is 0.
  • 0 real solutions (roots). This happens when (\(b^2- 4ac\)) is negative.