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Quadratic equations

Quadratic equations

  • A quadratic equation is an equation which contains only one variable but that variable must only be raised to powers that are positive whole numbers with a maximum value of \(2\), that is, of the form:
    \[y=ax^2+bx+c\]
  • Quadratic functions can be represented graphically by a curve called a parabola.

    Quadratic Equation

  • The parabola can take two forms:

    1. a minimum turning point when (\(a>0\))

    2. a maximum turning point when (\(a<0\))


  • In order to sketch a parabola you need to know:

    1. where it cuts the \(y\)-axis (\(y\)- intercept, i.e. \(x=0\))

    2. where it cuts the \(x\)-axis (if at all, i.e. \(y=0\))

    3. the coordinates of the turning point (axis of symmetry can be found \(x=-\frac{b}{2a}\)).

  • There are two techniques available to solve quadratic equations:

    1. Factorisation works on the principle that if the product of two expressions is zero then one or both of those expressions must be zero.
    2. If \[(x-a)(x-b) = 0\] then \[x-a=0 \quad \mbox{or} \quad x-b=0\] giving \[x=a \quad \mbox{or} \quad x=b\]
    3. Quadratic formula.

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