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Quadratic equations

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*}

    Quadratic_2
  • 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.