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MathematicsQuadratic Formula
Science and Mathematics Education Research Group
Supported by UBC Teaching and Learning Enhancement Fund 2012-2013
Department of Curriculum and Pedagogy
FACULTY OF EDUCATIONa place of mind
The Quadratic Formula
a
acbbx
2
42
Quadratic Formula Part I
Write the equation in the form
a
acb
a
bxa
ca
b
a
bxa
ca
b
a
bx
a
bxa
cxa
bxacbxax
4
4
2
42
44
22
22
2
2
2
22
22
02 cbxax 02 qpxa
Complete the square
Factor out a
a
acbq
a
bp
4
4,
2
2
Quadratic Formula Part II
Solve for x: 04
4
2
22
a
acb
a
bxa
a
acbbx
a
acb
a
bx
aa
acb
a
bx
a
acb
a
bxa
2
4
2
4
2
0,4
4
2
04
4
2
2
2
2
22
22
Divide by a
Square root both sides
The Quadratic Formula
Quadratic Formula I
What are the roots of the quadratic equation:
062 2 xx
4
481,
4
481
4
471,
4
471
2
3,2
3
2,2
3
2,2
x
x
x
x
x
E.
D.
C.
B.
A.
?
a
acbbx
2
42
Remember:
Solution
Answer: C
Justification: Before applying the quadratic formula, always check if the equation can be factored.
2
3,2
4
71
)2(2
)6)(2(4)1()1(
6,1,2,2
4
2
2
x
x
x
cbaa
acbbx
2,2
3
0232
062 2
x
xx
xxFactoring: Quadratic formula:
It is much faster to solve this question by factoring rather than using the quadratic formula.
Quadratic Formula II
What are the roots of the equation:
0132 xx
determined be CannotE.
D.
C.
B.
A.
2
53,
2
53
2
53
2
5
2
3,
2
5
2
3
2
5
2
3
x
x
x
x?
a
acbbx
2
42
Solution
Answer: B
Justification: First check if the quadratic can be factored. In this case it cannot, so apply the quadratic formula:
2
5
2
3,
2
5
2
3
2
53
)1(2
)1)(1(433
1,3,1,2
4
013
2
2
2
x
x
x
cbaa
acbbx
xx
Quadratic Formula III
Consider the part of the quadratic formula inside the square root. This is known as the discriminant, .
If a quadratic equation has 1 solution, what can be concluded about the discriminant?
0
11
0
0
0
E.
D.
C.
B.
A.
or
acb 42
acba
bx 4,
22
Solution
Answer: C
Justification: If a quadratic equation has 1 solution, then the quadratic formula must also give 1 value for x.
This can only happen in the case where:
If then the corresponding quadratic equation will have 1 solution.
a
acbbx
2
42
a
b
a
bx
22
0
,042 acb
Quadratic Formula IV
The discriminant of a quadratic is negative .
How many real solutions does the corresponding quadratic equation have?
0
A. 0, 1, or 2 real solutions
B. 0 or 1 real solution(s)
C. 2 real solutions
D. 1 real solution
E. 0 real solutions
Solution
Answer: E
Justification: If the quadratic formula were used to determine the solutions to a quadratic equation, we would get
If the discriminant is negative, we would get a negative value inside the square root. The quadratic equation therefore has no real solutions.
04,2
2
4
2
2
acba
bx
a
acbbx
Quadratic Formula V
How many real solutions does the following quadratic equation have?
edbedetermin CannotE.
solutions InfiniteD.
solutions real 0C.
solution real 1B.
solutions real 2A.
0131427 2 xx
Solution
Answer: A
Justification: Determine if the discriminant is greater than zero, equal to zero, or less than zero:
It is not necessary to calculate the exact value of the discriminant. It is clear from the above calculation that the discriminant will be the sum of 2 positive values. Since the discriminant is positive, the quadratic equation will have 2 solutions.
0
0)13)(27(414
)13)(27(4)14(
4
2
2
2
acb
Quadratic Formula VI
After using the quadratic formula to solve a quadratic equation, the following 2 solutions are found:
6
135x
What is the quadratic equation with this solution?
0156
0156
0153
01353
01353
2
2
2
2
2
xx
xx
xx
xx
xx
E.
D.
C.
B.
A.
Solution
Answer: C
Justification: Compare the solution with the quadratic formula:
From the above, we can see that -b = -5 and 2a = 6. After we find the values of a and b, we can use the discriminant to find c:
Since we now know a, b and c, the quadratic equation is therefore:
6
135x
a
acbbx
2
42
1
5,3,13)3(4513
42
2
c
bac
acb
0153 2 xx
5
5
b
b
3
62
a
a
Quadratic Formula VII
Consider a quadric function , where and .
Which of one of the following statements is true about function?
above the of NoneE.
all forD.
all forC.
all forB.
all forA.
xxf
xxf
xxf
xxf
0)(
0)(
0)(
0)(
cbxaxxf 2)( 0a042 acb
Solution
Answer: A
Justification: Since the discriminant is negative, the quadratic equation
has no solution. This means that the quadratic function never crosses the x-axis.
The quadratic function must either lie completely above or completely below the x-axis. Since we also know that , the quadratic opens upwards. Therefore, in order to never cross the x-axis, the graph must lie above the x-axis for all values of x.
cbxax 20
0)( 2 cbxaxxf
0a
0)( xf
Quadratic Formula VIII
Recall that a quadratic function
has a vertex at .
If and , which one of the following statements is true?
aa
b
4,
2
a
acb
a
bxaxf
4
4
2)(
22
0a 0
axis- xthe onvertex down, opens E.
axis-below xvertex down, opens D.
axis- xabovevertex down, opens C.
axis-below xvertex up, opens B.
axis- xabovevertex up, opens A.
)(
)(
)(
)(
)(
xf
xf
xf
xf
xf
Solution
Answer: D
Justification: If , then , so .
The parabola opens downwards.
When and the y-value of the vertex is negative. The vertex is therefore below the x-axis.
If the quadratic function opens downwards and the vertex is below the x-axis, it will never cross the x-axis and therefore has no solutions.
This agrees with the conclusion that when , the quadratic equation has no solutions.
0a 02
2
a
bxa
axf
4)(
0a 0 a4
0
Summary
The table below shows the connection between the discriminant and graphs of quadratic functions.
Number of solutions
+ + - 2 solutions - Opens up and vertex below x-axis
+ - + 0 solutions - Opens up and vertex above x-axis
- + +2 solutions - Opens down and vertex above x-
axis
- - - 0 solutions - Opens down and vertex below x-axis
+ 0 0 1 solution – Opens up and vertex on x-axis
- 0 0 1 solution – Opens down and vertex on x-axis
a a4
solutions 0
solution 1
solutions 2
,0
,0
,0