Quadratic Equation in One Unknown - (1.5-1.7)

Understand the relations between the discriminant of a quadratic equation and the nature of its roots

  • Theory

    In the quadratic equation \({\bf{a}}{{\bf{x}}^2} + {\bf{bx}} + {\bf{c}} = 0\), the discriminate \({\bf{\Delta }}\) decides how many roots the equation has \({\bf{\Delta }} = {{\bf{b}}^2} - 4{\bf{ac}}\;\left\{ {\begin{array}{*{20}{c}}{ > 0\;\;\;2\;real\;roots\;\;\;}\\{ = 0\;\;\;1\;real\;root\;\;\;\;}\\{ < 0\;no\;real\;roots\;}\end{array}} \right.\)

  • Examples

    Let k be a constant. Express the discriminant of the quadratic equation \(5{\bf{k}}{{\bf{x}}^2} + 24{\bf{x}} + 6 = 0\) in terms of k.
    If the equation has no real roots, find the range of values of k.

  • Solutions

    From the equation, \(\left\{ {\begin{array}{*{20}{c}}{a = 5k}\\{b = 24}\\{c = 6}\end{array}} \right.\)
    \({\rm{\Delta }} = {{\rm{b}}^2} - 4ac\)
    \({\rm{\Delta }} = {\left( {24} \right)^2} - 4\left( {5k} \right)\left( 6 \right)\)
    \({\rm{\Delta }} = 576 - 120{\rm{k}}\)
    the equation has no real root
    \(\therefore {\rm{\Delta }} < 0\)
    \({\rm{\Delta }} = {{\rm{b}}^2} - 4ac < 0\)
    \({\rm{k}} > \frac{{24}}{5}\)

solve problems involving quadratic equations

  • Examples

    In the figure, the area of rectangle ABCD is \(54{\rm{ c}}{{\rm{m}}^2}\). Find the value of x.

  • Solutions

    \({\rm{x}}\left( {{\rm{x}} - 3} \right) = 54\)
    \({{\rm{x}}^2} - 3x - 54 = 0\)
    \(\left( {{\rm{x}} - 9} \right)\left( {{\rm{x}} - 6} \right) = 0\)
    \({\rm{x}} = 9{\rm{\;or\;}}6\left( {{\rm{rejected}}} \right)\)
    \(\therefore {\rm{x}} = 9{\rm{\;cm}}\)

Understand the relations between the roots and coefficients and form quadratic equations using these relations

  • Theory

    If the root of the quadratic equation \({\rm{a}}{{\rm{x}}^2} + bx + c = 0\) are \({\rm{\alpha }}\) and \({\rm{\beta }}\),
    then, the sum of roots
    \({\rm{\alpha }} + {\rm{\beta }} = - \frac{{\rm{b}}}{{\rm{a}}}\)
    and, the product of roots
    \({\rm{\alpha \beta }} = \frac{{\rm{c}}}{{\rm{a}}}\)

  • Examples

    It is given that the sum of the roots of the quadratic equation \(8{{\rm{x}}^2} + {\rm{kx}}--56 = 0{\rm{\;}}\)is 6.
    (a) Find the value of k.
    (b) Solve the quadratic equation.

  • Solutions

    (a) ∵ Sum of roots = 6
    ∴ \(\begin{array}{l} - \frac{k}{8} = 6\\\quad k = \underline{\underline { - 48}} \end{array}\)
    (b) By substituting k = –48 into the equation, we have
    \(\begin{array}{c}8{x^2} + ( - 48)x - 56 = 0\\{x^2} - 6x - 7 = 0\\(x - 7)(x + 1) = 0\\x = \underline{\underline 7} \;\,\,{\rm{or}}\quad x = \underline{\underline { - 1}} \end{array}\)