Quadratic
Equations & Functions
2
x
Quadratic Equations have
(or some variable, squared) in
them and are equations.
x2 + 5x + 6 = 0
2
n – 7n = 18
2
2x = 11x + 40
p2 = 16
(x + 7)(x + 2) = 0
The key to solving quadratic
equations is the 0 property of
multiplication
 If the product of two
quantities is 0, then one of
those quantities must be
0.
So, if (x + 3)(x – 2) = 0,
then either x + 3 = 0
or x – 2 = 0
So, if (x + 3)(x – 2) = 0,
then either x + 3 = 0
or x – 2 = 0
This means either x = -3
or x = 2
If a quadratic equation is
factored, and says (__)(__) = 0:

The answers are the
opposite of the factors.
Solve
(x + 7)(x – 1) = 0
(x – 4)(x – 9) = 0
(x + 8)(x + 13) = 0
Solve
(x + 7)(x – 1) = 0
x = -7 or 1
(x – 4)(x – 9) = 0
x = 4 or 9
(x + 8)(x + 13) = 0
x = -8 or -13
What about these?
x(x + 9) = 0
(3x + 7)(x – 2) = 0
(5x – 1)(2x + 1) = 0
x(x + 9) = 0
Either x = 0 or x + 9 = 0
So x = 0 or x = -9
(3x + 7)(x – 2) = 0
Either 3x + 7 = 0 or x – 2 = 0
So
x=
-7/
3
or x = 2
(5x – 1)(2x + 1) = 0
Either 5x – 1 = 0 or 2x + 1 = 0
x = 1/5 or x = -½
When there are coefficients,
you can find the answers
quickly by making a fraction,
reading the factor backwards.
(9x – 13)(2x + 7) = 0
x=
13/
9
or x =
-7/
2
If the equation isn’t factored.
1. Write it so it says ___ = 0.
2. Factor.
3. Do the opposite of the
factors.
n2 – 7n = 18
n2 – 7n = 18
2
n
– 7n – 18 = 0
n2 – 7n = 18
– 7n – 18 = 0
(n – 9)(n + 2) = 0
2
n
n2 – 7n = 18
– 7n – 18 = 0
(n – 9)(n + 2) = 0
2
n
n = 9 or -2
x2 + 5x + 6 = 0
x2 + 5x + 6 = 0
Already says __ = 0
(x + 2)(x + 3) = 0
x2 + 5x + 6 = 0
Already says __ = 0
(x + 2)(x + 3) = 0
x = -2 or x = -3
2x2 = 11x + 40
2x2 = 11x + 40
2
2x
– 11x – 40 = 0
2x2 = 11x + 40
2
2x
– 11x – 40 = 0
(2x + 5)(x – 8) = 0
2x2 = 11x + 40
2
2x
– 11x – 40 = 0
(2x + 5)(x – 8) = 0
x = -2/5 or x = 8
2
p
= 16
2
p
= 16
p2 – 16 = 0
2
p
= 16
p2 – 16 = 0
(p + 4)(p – 4) = 0
p = 4 or -4
If a quadratic equation doesn’t
factor, there are many other
ways it could possibly be
solved. You’ll learn many of
these in Geometry and in
Advanced Algebra.
Quadratic Function
 Equation always has the
form f(x) = ax2 + bx + c
 The simplest quadratic
2
function is f(x) = x
Graph f(x) = x2
Like the absolute
value function, this
gives the same
answers for positive
and negative
numbers.
However it is curved at the
bottom rather than making a
straight V-shape.
This U-shaped graph is called
a parabola.
 Many things in the real
world form parabola (arch)
shapes.
Quadratic functions are used
particularly in problems
involving
 Movement and the force of
gravity
 Area
What we usually care about
with quadratic functions are
the roots.
 These are the places
where the function = 0
 They can also be called
“zeros” or “x-intercepts”.
The roots of
this quadratic
function are
-3 and 1.
This is the
same as the solutions to
-x – 2x + 3 = 0
Find the roots.
Find the roots.
Parabolas may have 2, 1, or 0
roots.
Find the roots of
y = (x + 3)(x – 9)
f(x) = (x – 5)(x + 3)
g(x) = (x +
2
2)
Just set the functions = 0
Find the roots of
y = (x + 3)(x – 9)
-3 and 9
f(x) = (x – 5)(x + 3)
5 and -3
2
g(x) = (x + 2)
-2
Find the roots:
f(x) = x2 + 11x + 28
y=
2
x
– 11x + 18
g(x) = x2 + 1x – 12
Find the roots:
f(x) = x2 + 11x + 28
(x + 7)(x + 4)
-7 and -4
2
y = x – 11x + 18
(x – 9)(x – 2)
9 and 2
g(x) = x2 + 1x – 12
(x + 4)(x – 3)
-4 and 3
Find the roots
y = x(x – 8)
Find the roots
y = x(x – 8)
0 and 8