Quadratic Equation in One Unknown  (1.11.4)
Solve quadratic equations by factor method

Examples
Solve the following quadratic equations by factor method.
\(64x^2 – 9 = 0\) 
Solutions
\(\begin{array}{c}64{x^2}  9 = 0\\(8x + 3)(8x  3) = 0\\8x + 3 = 0{\rm{ or 8}}x  3 = 0\\x = \underline{\underline {  \frac{3}{8}}} {\rm{ }}{\kern 1pt} {\rm{or }}x = \underline{\underline {\frac{3}{8}}} \end{array}\)
Form quadratic equations from given roots (The given roots are confined to real numbers. )

Examples
Given two roots of a quadratic equation are \(\frac{4}{5}\;\)and \(\frac{3}{4}\) respectively. Write the equation in the form \({\rm{a}}{{\rm{x}}^2} + bx + c = 0\)

Solutions
\(\left( {{\rm{x}}  \frac{4}{5}} \right)\left( {x  \frac{3}{4}} \right) = 0\)
\({{\rm{x}}^2}  \frac{4}{5}x  \frac{3}{4}x + \left( {\frac{4}{5} \cdot \frac{3}{4}} \right) = 0\)
\({{\rm{x}}^2}  \frac{{31}}{{20}}x + \frac{3}{5} = 0\)
\(20{{\rm{x}}^2}  31x + 12 = 0\)
Solve the equation \({\rm{a}}{{\rm{x}}^2} + {\rm{bx}} + {\rm{c}} = 0\) by plotting the graph of the parabola \({\rm{y}} = {\rm{a}}{{\rm{x}}^2} + {\rm{bx}} + {\rm{c}}\) and reading the xintercepts.

graph

Examples
Given the graph of the quadratic function \({\rm{y}} = \frac{1}{4}{x^2}  \frac{9}{4}\)
Solve \(\frac{1}{4}{x^2}  4 = 0\)

Solutions
From the given graph, it intersects at \(\left( {3,0} \right)\) and \(\left( {  3,0} \right)\).
\(\therefore \) The solution to \(\frac{1}{4}{x^2}  \frac{9}{4} = 0\) is 3 or 3.
solve quadratic equations by the quadratic formula

Theory
If \({\bf{a}}{{\bf{x}}^2} + {\bf{bx}} + {\bf{c}} = 0,\;\;\)Then \({\bf{x}} = \frac{{  {\bf{b}} \pm \sqrt {{{\bf{b}}^2}  4{\bf{ac}}} }}{{2{\bf{a}}}}\)

Examples
Solve \(\left( {3  {\rm{x}}} \right)\left( {{\rm{x}} + 3} \right) = \frac{{\left( {{\rm{x}} + 7} \right)\left( {{\rm{x}}  9} \right)}}{2}\)

Solutions
\(\left( {3  {\rm{x}}} \right)\left( {{\rm{x}} + 3} \right) = \frac{{\left( {{\rm{x}} + 7} \right)\left( {{\rm{x}}  9} \right)}}{2}\)
\(2\left( {9  {{\rm{x}}^2}} \right) = {x^2}  2x  63\)
\(18  2{{\rm{x}}^2} = {x^2}  2x  63\)
\(3{{\rm{x}}^2}  2x  81 = 0\)
\({\rm{x}} = \frac{{  \left( {  2} \right) \pm \sqrt {{{\left( {  2} \right)}^2}  4\left( 3 \right)\left( {  81} \right)} }}{{2\left( {  3} \right)}}\)
\({\rm{x}} = \frac{{2 \pm \sqrt {976} }}{6}\)
\({\rm{x}} = \frac{{1 \pm 2\sqrt {61} }}{3}\)