Quadratic equations

Quadratic equations: Solving quadratic equations

Quadratic equations

We are interested in the solutions of quadratic equations, these are equations of the form #ax^2+bx+c=0#. There are several ways to solve such an equation. We will start out by solving the simplest quadratic equation #x^2=\blue c#.

If #\blue c>0#, the equation has two solutions:

\[\begin{array}{rcl}
x^2 &=&\blue c \\
&\text{gives}&\\
x = \sqrt{\blue c} &\lor& x = -\sqrt{\blue c}
\end{array}\]

The symbol #\lor# means "or".

geogebra picture

If #\blue c=0#, the equation has one solution:

\[\begin{array}{rcl}
x^2 &=& \blue c \\
&\text{gives}&\\
x &=&0
\end{array}\]

geogebra picture

If #\blue c<0#, the equation has no solutions:

\[\begin{array}{rcl}
x^2 &=&\blue c
\end{array}\]

The graph on the right demonstrates that the graphs #y=\blue c# and #y=x^2# never intersect as long as #\blue c<0#.

geogebra picture

Solve #s# in the equation #s^2+33=38#.

Write:

#none# if there is no solution,
#s=s_1# if there is one solution,
#s=s_1\lor s=s_2# if there are two solutions,

with the correct values for #s_1# and #s_2#.

#s=-\sqrt{5}\lor s= \sqrt{5}#

#\begin{array}{rclcl}s^2+33&=&38\\&&\phantom{xxx}\blue{\text{the given equation}}\\ s^2&=&5\\&&\phantom{xxx}\blue{\text{constant terms to the right hand side}}\\ s=-\sqrt{5} & \lor &s= \sqrt{5}\\&&\phantom{xxx}\blue{\text{according to the theory}}\end {array}#

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