### Quadratic Equation in One Unknown - (1.1-1.4)

#### Solve quadratic equations by factor method

• Solve the following quadratic equations by factor method.
$$64x^2 – 9 = 0$$

• $$\begin{array}{c}64{x^2} - 9 = 0\\(8x + 3)(8x - 3) = 0\\8x + 3 = 0{\rm{ or 8}}x - 3 = 0\\x = \underline{\underline { - \frac{3}{8}}} {\rm{ }}{\kern 1pt} {\rm{or }}x = \underline{\underline {\frac{3}{8}}} \end{array}$$

#### Form quadratic equations from given roots (The given roots are confined to real numbers. )

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

• $$\left( {{\rm{x}} - \frac{4}{5}} \right)\left( {x - \frac{3}{4}} \right) = 0$$
$${{\rm{x}}^2} - \frac{4}{5}x - \frac{3}{4}x + \left( {\frac{4}{5} \cdot \frac{3}{4}} \right) = 0$$
$${{\rm{x}}^2} - \frac{{31}}{{20}}x + \frac{3}{5} = 0$$
$$20{{\rm{x}}^2} - 31x + 12 = 0$$

#### 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 x-intercepts.

• Given the graph of the quadratic function $${\rm{y}} = \frac{1}{4}{x^2} - \frac{9}{4}$$
Solve $$\frac{1}{4}{x^2} - 4 = 0$$

• 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.

• If $${\bf{a}}{{\bf{x}}^2} + {\bf{bx}} + {\bf{c}} = 0,\;\;$$Then $${\bf{x}} = \frac{{ - {\bf{b}} \pm \sqrt {{{\bf{b}}^2} - 4{\bf{ac}}} }}{{2{\bf{a}}}}$$

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

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