
Solving Quadratic Equation
1. The equation is of the form a(x − h)^2 + k = 0
Corresponding parabola or quadratic function: y = a(x − h)^2 + k
Solutions are xintercepts of this parabola
don’t forget ± sign
• Solutions are
• Simplify and write as 2 separate numbers if − k/a is a
perfect square
2. The equation is not in the above form.
• If the equation is not in the form ax^2 + bx + c = 0,
then bring every term on one
side of “=”, foil (if necessary) and simplify to ax^2 + bx + c = 0
Corresponding parabola or quadratic function: y = ax^2 + bx + c
Solutions are xintercepts of this parabola
• The solution is
Simplify and write as 2 separate numbers if b^2 − 4ac is a
perfect square
You will get:
Discriminant

Type of solution (*
if p, q, r or a, b, c are integers) 
Graphically 
positive perfect
square
not a perfect square 
2 real solutions 2
rational solution*
2 real solution with radicals
conjugate to each other 
2 xintercepts
parabola crosses xaxis twice 
zero 
1 rational solution* 
only 1 xintercept
parabola just touches x axis 
negative 
2 complex solutions
conjugate to each other 
no xintercept
parabola does not intersect x axis 
Graphing Quadratic Function: Vertical Parabola
1. The function is in the form y = a(x − h)^2 + k
• Plot the points (h, k), (h + 1, k + a) and (h − 1, k + a)
• Draw the parabola through these points.
2. The function is in the form y = ax^2 + bx + c
• Plot the points:
yintercept  (0, c), Point of symmetry(or
plug in x = − b/2a into y = ax^2 + bx + c to get ycoordinate of vertex)
• Draw the parabola through these points.
• a > 0  parabola opens up (smilie) with minimum at the
vertex
• a < 0  parabola opens down (frownie) with maximum at the vertex

y = a(x − h)^2 + k 
y = ax^2 + bx + c 
vertex 
(h, k) 

axis 
x = h 

symmetric points* 
(h + 1, k + a) and (h − 1, k + a) 
(0, c) and

yintercept 
(0, ah^2 + k) 
(0, c) 
xintercept 


none 
if k > 0 
if b^2 < 4ac 
one 
if k = 0 then (h, 0) 
if b^2 = 4ac then

two 
if k < 0 then 
if b^2 > 4ac then

* These points are on opposite sides of the axis, at equal
distance from the axis and are at
the same height i.e. they have the same ycoordinate.
For horizontal parabola:
x = a(y − k)^2 + h  plot (h, k), (h + a, k + 1) and (h +
a, k − 1) and draw the parabola
x = ay^2 + by + c  plot (c, 0),and draw the
parabola
a > 0  parabola opens right, a < 0  parabola opens left
