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SOLVING QUADRATICS
Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources
SOLVING QUADRATICS
•General Form:
cbxaxy 2
•Where a, b and c are constants
02 cbxax
To solve a quadratic equation, the
equation must be expressed in the form:
That is, all variables and constants must be
on one side of the equals sign with zero on
the other.
02 cbxax
Methods for Solving Quadratic equations
•Method 2 : Quadratic Formula
a
acbbx
2
42
•Method 1 : Factorisation
•Method 4 : Graphic Calculator
•Method 3 : Completing the Square
•Method 1 : Factorisation
Example 1 : 020122 xx
020122 xx
0210 xx
0)2(0)10( xx or either
10x 2xor
Step 1: Equation
in the right form
Step 2: Factorise
Step 3: Separate
the two parts of
the product
Step 4: Solve
each equation
Example 1 : 020122 xx
020122 xx
0210 xx
02010 xx or either
10x 2xor
Some points to note
Need to remember your
factorisation skills
The equation is expressed as the product of two factors being equal to zero, therefore, one (or both) of the factors must be zero.
To check your answers are correct you can substitute them
one at a time into the original equation.
Example 1 : 020122 xx
10x
2x
Checking solutions
0
20120100
20)10(12102
0
20244
20)2(1222
•Method 1 : Factorisation
Example 2 : 2832 xx
02832 xx
047 xx
0407 xx or either
7x 4xor
Step 1: Equation
in the right form
Step 2: Factorise
Step 3: Separate
the two parts of
the product
Step 4: Solve
each equation
•Method 2 : Quadratic Formula
Example 1 : 0582 xxa
acbbx
2
42
)1(2
)5)(1(4)8(8 2 x
2
20648 x
2
848x
2
2128x
a = 1 b = – 8 and c = – 5
Step 1: Determine the values of a, b and c
Step 2: Substitute the
values of a, b and c
Step 3: Simplify
See next slideThis step is optional. Students need to have covered surds
2
2128x
2
2142 x
214x
58.0214
58.8214
x
Exact answer
Decimal
approximation
Note that 2 is a factor
of the numerator
Once 2 is factored out, it
can be cancelled with the
2 in the denominator
•Method 2 : Quadratic Formula
Example 2 : 0283 2 xxa
acbbx
2
42
)3(2
)2)(3(488 2 x
6
24648 x
6
408x
6
1028x
a = 3 b = 8 and c = 2
Step 1: Determine the values of a, b and c
Step 2: Substitute the
values of a, b and c
Step 3: Simplify
See next slide
This step is optional. Students need to have covered surds
6
1042 x
3
104x
39.23
104
28.03
104
x
Exact answer
Decimal
approximation
Note that 2 is a factor
of the numerator
Once 2 is factored out, it
can be cancelled with the
6 in the denominator
6
1028x
•Method 2 : Quadratic Formula
Example 3 : 0675 2 xxa
acbbx
2
42
)5(2
)6)(5(4)7(7 2 x
10
120497 x
10
717 x
a = 5 b = – 7 and c = 6
Step 1: Determine the values of a, b and c
Step 2: Substitute the
values of a, b and c
Step 3: Simplify
Problem - you cannot
find the square root of
a negative numberNo real Solutions
That was a lot of work to find
that there was no solution!
It would be useful to be able to
“test” the equation before we
start.
For this we use the
DISCRIMINANT.
The Discriminant
The discriminant is a quick way to check how many real solutions exist for a given quadratic equation.
acb 42
As shown above the symbol for the
discriminant is and it is
calculated using .acb 42
Summary of Results using Discriminant
> 0The equation has two
real solutions
acb 42
= 0 The equation has one
real solutions
< 0The equation has no
real solutions
Relating the Discriminant to graphs
> 0
acb 42
The graph cuts the x-axis in two places. These are the two real solutions to the quadratic equation.
y)
x
y
x
y
x
Relating the Discriminant to graphs
acb 42
= 0
These graphs have their turning point on the x-axis and hence there is only the one solution.
y)
x
y
x
y
x
Relating the Discriminant to graphs
acb 42
< 0
There are no solutions in this case because the graphs do not intersect with the x-axis.
y)
x
y
x
y
x
•Method 3: Completing the Square Technique
0362 xx
039)3( 2 x
06)3( 2 x
Example 1 :
This value
is half b
Subtract the square of the number in the bracket
6)3( 2 x
63 x
63 x
Add 6 to both sides
Take the square root of both sides
Subtract 3 from both sides
•Method 3: Completing the Square Technique
This result gives us the exact answers.
63 x
or63 x 63 xUse your calculator to find decimal
approximations accurate to two decimal places.
45.555.0 xx or
•Method 3: Completing the Square Technique
0852 xx
084
25)
2
5( 2 x
04
57)
2
5( 2 x
Example 2 :
4
57)
2
5( 2 x
2
575.2 x
2
575 x
Take the square root of both sides
Subtract 2.5 from both sides
This value
is half b
Subtract the square of the number in the bracket
Add to both sides4
57
Example 3: Solve 01072 xx
01072 xx
0]25.2)5.3[( 2 x
01025.12)5.3( x
Factor out the
coefficient of x²
025.2)5.3( 2 x
This value
is half b
Subtract the square of the number in the bracket
25.2)5.3( 2 x
25.2)5.3( 2 xSee next slide
25.2)5.3( 2 x
25.25.3 x
25.25.3 x
or i.e. 25.25.3 x 25.25.3 x
•Method 4 Graphic Calculator
Graph the function and find the value of the x-intercepts
Use a solver function for a
polynomial of degree 2
