Solving Quadratic Equations Graphically and Algebraically
A quadratic equation is an equation that can be simplified to follow the pattern:
y = a Â· x^{2} + bÂ· x + c,
where the letter x represents the input, the letter y represents the value of the output and
the letters a, b and c are all numbers. Sometimes the numbers a, b and c are referred to as
coefficients.
Solving a Quadratic Equation
Just as you did for linear and power equations, you solve a quadratic equation when you
have been given a yvalue and need to find all of the corresponding xvalues. For
example, if you had been given the quadratic equation:
y = x^{2} + 8 Â· x +10,
and the yvalue,
y = 30,
then solving the quadratic equation would mean finding all of the numerical values of x
that work when you plug them into the equation:
x^{2} + 8 Â· x +10 = 30.
Note that solving this quadratic equation is the same as solving the quadratic equation:
x^{2} + 8 Â· x +10  30 = 30  30 (Subtract 30 from each side)
x^{2} + 8 Â· x  20 = 0 (Simplify)
Solving the quadratic equation
x^{2} + 8 Â· x  20 = 0 will give exactly the same values for x
that solving the original quadratic equation,
x^{2} + 8 Â· x +10 = 30, will give.
The advantage of manipulating the quadratic equation to reduce one side of the equation
to zero before attempting to find any values of x is that this manipulation creates a new
quadratic equation that can be solved using some fairly standard techniques and formulas.
Solving a Quadratic Function Graphically Using a Graphing Calculator
When you are trying to solve a quadratic equation of the form:
a Â· x^{2} + b Â· x + c = 0,
the solutions are the xcoordinates of the points where the graph of the quadratic equation
y = aÂ· x^{2} + bÂ· x + c cuts the horizontal axis.
From a graphical point of view, the solutions of the manipulated quadratic formula:
x^{2} + 8 Â· x  20 = 0,
are the xvalues where the graph of
y = x^{2} + 8 Â· x  20 cuts the horizontal xaxis. (These
points, x = 2 and x = 10, are shown in Figure 1).
Figure 1: Graphical representation of the
solutions of a quadratic equation.
There may be zero, one or two places where
the graph of the quadratic equation
y = a Â· x^{2} + bÂ· x + c cuts the horizontal xaxis
(see Figure 2, below). This means that the
quadratic equation:
a Â· x^{2} + b Â· x + c = 0,
can have zero, one or two solutions.
You can determine the number of solutions
that a quadratic equations has by calculating
the discriminant,Δ. The discriminant is equal
to:
Δ = b^{2}  4 Â· a Â· c .
The sign of the discriminant tells you the
number of solutions that exist for the
quadratic equation,
a Â· x^{2} + b Â· x + c = 0.
Figure 2: (a) A quadratic equation with zero solutions. (b) A quadratic equation with exactly one solution. (c) A quadratic
equation with exactly two solutions.
Sign of discriminant, Δ = b^{2}  4
Â· a Â· c 
Number of solutions of quadratic equation,
a Â· x^{2} + b Â· x + c = 0 
Negative () 
Zero solutions 
Zero (0) 
Exactly one solution 
Positive (+) 
Exactly two solutions.

Table 1: Determining the number of solutions using the sign of the discriminant.
