A quadratic equation is defined as an
equation in which one or more of the terms is squared but raised to no
higher power. The general form is ax^{2} + bx
+ c = 0, where a, b and c are constants.
In Algebra and Geometry, we learned how
to solve linear  quadratic systems algebraically and graphically. With our new found knowledge of quadratics, we are now
ready to attack problems that cannot be solved by factoring,
and problems with no real solutions.
The familiar linearquadratic system:
(where the quadratic is in one variable) 
Remember that linearquadratic systems
of this type can result in three graphical situations such as:



The equations
will intersect in two locations. Two real solutions. 
The equations
will intersect in one location. One real solution. 
The equations will not
intersect.
No real solutions. 
Keep these images in mind as we proceed to solve these
linearquadratic
systems algebraically.

When we studied these systems in
Algebra, we encountered situations that
could be solved by factoring, such as this
first example. 
Solve this system of equations algebraically:
y = x^{2} 
x  6
(quadratic equation in
one variable of
form
y =
ax^{2}
+
bx
+
c
)
y = 2x  2
(linear
equation of form y = mx + b)
Substitute
from the linear equation into the quadratic
equation and solve.
y
= x^{2}
 x  6
2x  2
= x^{2}
 x  6
2x
= x^{2}  x  4
0 = x^{2}
 3x  4
0 =(x
 4)(x + 1)
x 
4 = 0 x + 1 =0
x = 4 x = 1 
Find the
yvalues by substituting each value of
x into the linear
equation.
y
= 2(4)
 2 = 6
POINT (4,6)
y = 2(1)
 2 = 4
POINT (1,4)

See how to use your
TI83+/84+ graphing
calculator with
quadraticlinear systems.
Click calculator. 


There are
2 "possible" solutions for the system: (4,6)
and (1,4)
Check each in both equations.
y = x^{2} 
x  6
6 =
(4)^{2}  4  6 = 6
checks
y = 2x  2
6 =
2(4)  2 = 6
checks
y = x^{2}  x  6
4
= (1)^{2}  (1)  6 = 4
checks
y = 2x  2
4
= 2(1)  2 = 4
checks
Answer:



With our new found knowledge of quadratics, we are now
ready to attack problems that cannot be solved by factoring,
and/or problems with no real solutions (such as this second
example). 
Solve this
system of equations algebraically:
y = x^{2}  2x +
1 (quadratic
equation in one variable of form
y =
ax^{2}
+ bx
+ c
)
y = x  3
(linear equation of form y
= mx + b)
Substitute from the linear equation into the quadratic
equation.
y
= x^{2}
 2x + 1
x 
3 = x^{2}
 2x + 1
0 = x^{2}
 3x + 4
Use quadratic formula:
No real solutions. 
Find
the yvalues by substituting each value of x
into the linear
equation.
POINT
POINT 
There are 2 "possible" solutions. Check each in both
equations.
y = x^{2}  2x +
1
y =
x  3

y = x^{2}  2x + 1
y =
x  3

Answer:
There are no real solutions.
The answers are complex numbers,
which are not graphed in the Cartesian coordinate plane.

Other
linear  quadratic systems:
(where the quadratic is in two variables) 
Quadratics in two variables look like x^{2}
+ y^{2} = 16 where two variables are squared.
Solve this
system of equations algebraically:
x^{2}
+ y^{2} = 25 (quadratic
equation of a circle center (0,0), radius 5)
4y =
3x
(linear equation)
Substitute
from the linear equation into the quadratic
equation and solve.

Find the
yvalues by substituting each value of
x into the linear
equation.
Answer:

There are
2 "possible" solutions for the system: (4,3)
and (4,3)
Check
each in both equations.
x^{2} + y^{2}
= 25
4^{2} + 3^{2}
= 25
16
+ 9 = 25
25 = 25
checks
4y = 3x
4(3)=3(4)
12 = 12
checks
x^{2} + y^{2}
= 25
(4)^{2} + (3)^{2}
= 25
16
+ 9 = 25
25 = 25
checks
4y = 3x
4(3)=3(4)
12 = 12
checks

Solve this
system of equations algebraically:
x^{2}
+ y^{2} = 26 (quadratic
equation)
x
 y =
6
(linear equation)
Substitute
from the linear equation into the quadratic
equation and solve.

Find the
yvalues by substituting each value of
x into the linear
equation.
Answer:

There are
2 "possible" solutions for the system: (4,3)
and (4,3)
Check
each in both equations.
x^{2} + y^{2}
= 26
5^{2} + (1)^{2}
= 26
25
+ 1 = 26
26 = 26
checks
x  y = 6
5  (1) = 6
6 = 6
checks
x^{2} + y^{2}
= 26
1^{2} + (5)^{2}
= 26
1
+ 25 = 26
26 = 26
checks
x  y = 6
1  (5) = 6
6 = 6
checks 
