Section 1.4 - Roots of Quadratic Equations
Specific Curriculum Outcomes covered
B10
Derive and apply the Quadratic Formula
B11 (Math 3205 only)
Analyze the Quadratic Formula to connect its components
to the graphs of Quadratic Functions
C22
Solve Quadratic Equations
C15
Relate the nature of the roots of quadratic equations and the x-intercepts of
the graphs of corresponding functions
A9
Represent non-real roots of quadratic equations as complex numbers
A3
Demonstrate an understanding of the role of irrational numbers in applications
A4
Demonstrate an understanding of the nature of the roots of quadratic
equations
Assumed Prior Knowledge
G
G
G
Factoring and expanding quadratic expressions (see diagnostic worksheet for
Section 1.3)
Using completing the square to create quadratic expressions ( See Focus D).
Solving equations
Page -1-
Focus F - Roots of Quadratic Equations
Read the introduction to the section, top of text p.41.
GLOSSARY TERMS (see text p. 41-42):
Quadratic Equation, Root of a Quadratic Equation, Zeros of a Function
Note: Be sure you understand the difference between a Quadratic Equation and a
Quadratic Function.
Function:
y = ax2+bx+c, a…0
Equation:
ax2+bx+c = 0, a…0 This is the special case of the function when
the value of y is 0
The text outlines four methods of solving quadratic equations (that is, finding
roots):
Method 1 -
as suggested above, graph the corresponding function and find the xintercepts
Method 2 and Method 3 -
these two methods are both based on factoring.
There are also other methods of factoring, such as
the “trial-and-error” and “borrow-and-pay back”
methods. Use the method with which you are most
comfortable; however, it is highly impractical to
factor many quadratics using algebra tiles, so you
will need another method other than that as well.
You also did similar work on solving Quadratic
Equations by Factoring in Math 1204.
Page -2-
Page -3-
Method 4 - Solving by Completing the Square. An extra example is provided below.
Solve:
Solution:
2x2 + 16x - 18 = 0
x2 + 8x - 9 = 0
x2 + 8x = 9
x2 + 8x + 16 + = 9+16
(x + 4)2 = 25
x + 4 = ±5 L
x = -4 + 5
x=1
or
or
x = -4 - 5
x = -9
You should check your answers by substituting them in the original equation:
2(1)2 + 16(1) - 18 = 0
Check x = -9:
2(-9)2 + 16(-9) Check x = 1:
18 = 0
2 + 16 -18 = 0
162 - 144 - 18 =
0
0=0
0=0
Both solutions “balance” the equation.
You should also note that a linear equation has only one root, while a quadratic
equation generally has two (can you think of situations when there might be fewer?
Hint: Think about the graph of a quadratic function)
Solving Rational Equations (see text p.46 #16, 17)
Review of solving an equation with fractional coefficients:
Example:
1
1 5
x+ =
2
3 6
LCD = (2)(3) = 6
1
1
 5
6 x +  = 6 
2
 6
3
Multiply both sides by the LCD
1 
 1
 5
6 x + 6  = 6 
2 
 3
 6
Simplify
3x + 2 = 5
3x = 3
x=1
Solve the resulting equation
Page -4-
4
3x2 + 2x
A Rational Expression is the quotient of two polynomials e.g.
and
.
x+2
4x − 3
Think of a rational expression as a fraction that contains variables.
A Rational Equation is an equation in which the variable occurs in the denominator;
its terms are rational expressions.
The skills required for solving a rational equation initially are the same as those in
the example above; however, once the fractions (rational expressions) have been
eliminated, the resulting equation may require other skills such as those for solving
quadratic equations.
Example:
4
= x+ 1
x−2
4 
( x − 2)
 = (x − 2)( x + 1)
 x − 2
4 = x2 − x − 2
x2 − x − 6 = 0
(x − 3)(x + 2) = 0
x − 3 = 0, or x + 2 = 0
LCD is x-2
Multiply both sides by the LCD
Eliminate the denominators and simplify
the equation.
This equation is quadratic and, in this
case, can be solved by factoring.
x = 3 or x = − 2
Example:
2
3
+ =2
x−3 x
3
 2
x(x − 3)
+  = 2x(x − 3)
 x − 3 x
LCD is x(x-3)
Multiply both sides by the LCD
 2 
 3
x(x − 3)
 + x(x − 3)  = 2x(x − 3)
 x − 3
 x
Eliminate the denominators and
simplify the equation
2x + 3x − 9 = 2x 2 − 6x
2x 2 − 11x + 9 = 0
(2x − 9)(x − 1) = 0
2x − 9 = 0, or x − 1 = 0
9
x = , or x = 1
2
Page -5-
Solving Problems Involving Speed, Distance, and Time (see text p. 47 #18-22)
Before tackling these questions, you should understand the following formula from
Physics:
distance = rate × time
or
d = rt
Be able to rearrange it e.g. r =
Example:
d
t
If a car travels 300km in 5 hours, then its average speed, r, can
be calculated as
Example:
300km
= 60km / h
5h
If a car travels 200km in t hours, then we can write the
expression r =
200
t
If you understand these examples, then you should be ready to follow the next
example.
EXAMPLE: A boat travels 120km from its home port to a fishing area. On the way
back, the boat travels 10km/h slower because it has a full load. If the
return trip takes 2h longer, what is the total travelling time for each
trip?
Solution
Let t represent the time on the initial trip out
Therefore, t+2 is the time for the return trip in.
Distance (km)
Speed (km/h)
Time (h)
Out
120
120
t
t
In
120
120
t+2
t+2
The difference between the speed out and the speed in is 10km/h. Therefore, we
write and solve the following equation:
Page -6-
120 120
−
= 10
t
t+ 2
 120 120 
t(t + 2)
−
 = 10t(t + 2)
 t
t + 2
120(t + 2) − 120t = 10t(t + 2)
Speed out - Speed in = 10
Multiply both sides by the LCD t(t+2)
Simplify and solve
2
120t + 240 − 120t = 10t + 20t
10t2 + 20t − 240 = 0
t2 + 2t − 24 = 0
(t + 6)(t − 4) = 0
t + 6 = 0 or t − 4 = 0
t = −6
or
t= 4
Do both of these answers make sense
for this problem? Explain.
The time to go out is therefore 4 hours. The time to come back in is 6 hours (2 hours
longer than the time to go out).
What you should have learned from this Focus:
G
G
G
G
G
How to solve a Quadratic Equation by Graphing, Factoring, and Completing
the Square
Finding the Zeros of a Quadratic Function is equivalent to finding the roots
of the corresponding Quadratic Equation (see text p. 42 and question # 10,
p. 43)
How to solve problems involving Quadratic Equations
How to create a Quadratic Function given the zeros (see #12, p. 46)
How to solve rational equations
Page -7-
Investigation 5 -
Developing and Analyzing the
Quadratic Formula
For any quadratic equation written in the form ax2 + bx + c = 0, there is a formula
that will allow you to find the roots very quickly.
Quadratic Formula:
Examples:
a)
x=
− (b) ± b2 − 4ac
2a
Solve each of the following
2x2 + 11x + 9 = 0
b)
a=2, b=11, c=9
Therefore,
Rearrange into general form
first:
-4x2 - 2x + 3 = 0
− (11) ± (11) 2 − 4(2)(9)
x=
2(2)
x=
x=
x=
x=
x=
− 11 ± 121 − 72
4
− 11 ± 49
4
− 11 ± 7
4
− 11 + 7
− 11 − 7
or x =
4
4
9
−1
or
x= −
2
a=-4, b = -2, c = 3
Therefore,
− (-2) ± (-2) 2 − 4(-4)(3)
x=
2(-4)
x=
x=
x=
x=
x=
NOTE:
-4x2 - 2x = -3
2 ± 4 + 48
-8
2 ± 52
-8
2 ± 2 13
-8
2 + 2 13
or
-8
-1 - 13
or
4
2 − 2 13
-8
-1 + 13
x=
4
x=
Math 3205 students will show how the quadratic formula is obtained by
using completing the square in #23, text p. 48
Page -8-
Text p. 48, #26.
This question has you prove the following formula:
Give a quadratic function in the form y = ax2 + bx + c, the x- coordinate of the
vertex is − b . Here is an example of how it can be a useful result:
2a
Function y = 2x2 - 8x + 4
x-coordinate of vertex is − -8 = 2
2(2)
Substitute this value into the function to get the y-coordinate:
y = 2(2)2 - 8(2) + 4 = -4
Therefore, the vertex is (2, -4)
**Incidentally, the axis of symmetry is x = 2, Why?
Text p.49, # 29
This question introduces the concept of imaginary numbers. The following is some
extra background information and practice exercises dealing with imaginary numbers
in case you need it:
The imaginary unit is i = − 1 . This is defined as such because no real
number can be the square root of a negative real number
e.g. 25 = 5, because 5 × 5 = 25
but − 25 ≠ − 5 or 5.
We can get the following useful result by squaring both sides of our imaginary
unit equation
i = -1
(i) 2 =
(
−1
)
2
i 2 = −1
Example 1:
Example 2:
− 25 =
= -25
i 2 × 25 = i 25 = 5i since 5i × 5i = 25i2 = 25(-1)
− 9 = 3i
Page -9-
Example 3:
− 32 =
i 2 × 32 = i 32 = 4i 2
Page -10-
Practice Set I - Find the square roots, in EXACT form, of each of the
following numbers: (answers at end of next page)
a) -4
b) -16
c) -36
d) -18
e) -8
f) -27
g) -125
Adding and subtracting complex numbers:
This is done in a manner similar to that for polynomial expressions - you add like
terms
=
(2 + 4) + (3i - 5i)
e.g. (2 + 3i) + (4 - 5i)
=
6 - 2i
=
(2 + 3i) + (-4 + 5i)
e.g. (2 + 3i) - (4 - 5i)
=
(2 - 4) + (3i + 5i)
=
-2 + 8i
Multiplying Complex Numbers
Again, this is done in a manner similar to multiplying binomials, but the simplification
can be carried a little further:
8 - 10i + 12i -15i 2
e.g. (2 + 3i)(4 - 5i) =
=
8 + 2i - 15i 2
=
8 + 2i - 15(-1)
=
8 + 2i + 15
=
23 + 2i
Practice Set 2 - Perform the indicated operation and simplify as far as possible:
a) (2 + 3i) + (5 + 4i)
d) (4 - 2i) - (-6 + 3i)
b) (-3 - 4i) + (5 - 2i)
e) (-2 - 6i) - (8 - 3i)
c) (5 - 3i)(9 + i)
f) (7 + 8i) - (5 + i)
g) (2 - 3i)(4 + 7i)
h) 3i(4 - 7i)
i) ( 6 + 12i) 
2 3 3 7 
− i +  + i
5 4  5 4 
j) 
m) (5 - 2i)(5 + 2i)


o)  −
5 2  5 2 
− i  − + i
4 7  4 7 
(
n) (
)(
k) 1 + i 5 3 − i 5
)
2 + i 3 )( 2 − i 3 )
p) (a + bi)(a - bi)
Page -11-
1 1 
− i
2 3 
l) (9 - 2i)(9 + 2i)
What you should have learned from this Investigation:
G
G
G
G
Quadratic Formula
− (b) ± b2 − 4ac
x=
2a
for equation in general
form
ax2 + bx + c = 0.
How to use the Quadratic Formula to find roots of quadratic equations and
x-intercepts of quadratic functions
(MATH 3205 only)
How to use completing the square to prove the
Quadratic Formula (see text, p.48 #23)
For any quadratic function in general form y = ax2 + bx + c, the x-coordinate
of the vertex is −
b
. Substitute this value into the function to get the y2a
coordinate (see text p.48, #26-27)
G
G
G
The Imaginary Unit, i, is defined asi = − 1 (see text, p. 49 #29)
An Imaginary or Complex Number can be represented by the expression a +
bi, where a and b are real numbers and i = − 1 (see text, p. 49 #29)
How to use imaginary numbers when finding the roots of quadratic equations
Practice Set I -Answers:
a) 2i
b) 4i
c) 6i
Practice Set II - Answers
b) 2 - 6i
a) 7 + 7i
2i
i) 7 + 4i
h) 21 + 12i
o) 1289/784
d) 3i /2 e) 2i /2 f) 3i /3 g) 5i /5
j) 1 + i
c) 48 -22i
d) 10 -5i
e) -10 -3i
f) 2 + 7i
k) 8 + 4i /5
l) 85
m) 29
n) 5
p) a2 + b2
Page -12-
g) 29 +
Focus G -
Using the Quadratic Formula to Solve
Problems
Read the Focus, p. 50 - 52.
What you should have learned from this Focus:
G
G
How to set up a quadratic equation in a problem solving situation.
How to interpret the solution to the equation in context (that is, identify
roots that do not make sense for the problem)
Page -13-
Investigation 6 -
Glossary Terms:
The Number of Roots of a
Quadratic Equation
Discriminant, Double Root
2
When considering the quadratic formula, x = − (b) ± b − 4ac , the portion under the
2a
2
square root sign (b - 4ac) determines the number and type of roots you will obtain.
Solve each of the following equations using the quadratic formula and complete
the indicated information.
1.
2x2 - 9x + 4 = 0
2.
x2 - 2x + 1 = 0
3.
2x2 - 2x + 1 = 0
# of roots?
# of roots?
# of roots?
Type of root(s)?
9 Real 9
Complex
Discriminant?
9 Positive
9 Zero
9 Negative
Graph of corresponding
function:
Type of root(s)?
9 Real 9
Complex
Discriminant?
9 Positive
9 Zero
9 Negative
Graph of corresponding
function:
Type of root(s)?
9 Real 9
Complex
Discriminant?
9 Positive
9 Zero
9 Negative
Graph of corresponding
function:
Page -14-
Complete the Investigation, noting the same information for each equation as in the
chart above.
What you should have learned from this Investigation:
G
G
G
G
G
G
The Discriminant is b2 - 4ac from the quadratic formula.
There are three cases for the number of roots /x-intercepts that
correspond to the value of the Discriminant:
•
If b2 - 4ac > 0, then there are two unequal real roots (or two xintercepts)
•
If b2 - 4ac = 0, then there is one real root (or one x-intercept)
•
If b2 - 4ac < 0, then are two unequal complex roots (or no real roots,
or no x-intercepts)
Functions of the form y = (x - q)2 have only one x-intercept, corresponding
to a discriminant value of 0 (see text, p.55, #51).
Functions of the form y = (x - r)(x - s), r … s, have two x-intercepts and
correspond to a positive Discriminant (see text, p.56, #52)
How to predict the number and type of roots of a quadratic equation by
examining the graph of the corresponding function
Given a function, use the graph and what you know about translations and
reflections to predict the number of x-intercepts for related functions
Page -15-
Investigation 7 -
Sum and Product of Roots
(Math 3205 only)
Complete the Investigation.
What you should have learned from this Investigation:
G
G
For a quadratic equation written in the form x2 + bx + c = 0, the sum of the
roots is -b and the product is c (see next point for more general result).
For a quadratic equation written in the form ax2 + bx + c = 0, the sum of the
roots is −
G
b
c
and the product is (see text, p.58, #67).
a
a
Given the roots, a quadratic equation may be written in the form x2 - (sum of
the roots)x + (product of the roots) = 0
e.g. If sum of the roots is 6 and the product is 3, then the quadratic
equation is x2 - 6x + 3 = 0
e.g. If the roots are 1 +
product is
(1+ 2)(1− 2)= − 1
2 and 1 − 2 , then the sum is 2 and the
. Therefore, the equation is
x2 - 2x - 1 = 0.
Extra Practice!!!!! Incl questions with complex roots.
idea - if an eq. has roots 2, 3 then another way is (x-2)(x-3) = 0 and mult the
factors. If the roots are (1+ /3) and 1 - /3, the factors are ....... Which method is
quicker - mult. the factors of sum/prod. method?
Page -16-
Extra Practice for Section 1.4
Focus F
1.
Estimate the roots of the given equation from the graph, then verify your
answers by substituting them in the equation.
a)
x2 + 8x + 7 = 0
b)
-3x2 = -6x
4
8
4
4
-8
-4
-4
-4
-8
c)
4
8
12
-10
-20
-30
-40
-50
2.
Put each function in transformational form, accurately graph it, and estimate
the zeros from the graph:
a) y = 2x2 - 8
b) y = x2 - 6x + 8
c) f(t) = 2t2 - 6t + 5
d) f(x) = -0.5x2 + 2x + 3 e) f(a) = 6 + 3a - 3a2
3.
Solve by factoring:
a) 2x2 - 8 = 0
d) 9x2 + 30x + 25 = 0
g) 4x2 - 11x -3 = 0
4.
Solve each equation in # 3 by completing the square.
b) x2 - 6x = -8
e) -14x2 - 6x = 0
h) 28z2 = -3z + 1
Page -17-
c) 0 = 6 + 3a - 3a2
f) 3x2 - 4 = 11x
5.
Solve each equation by completing the square.
a) x2 - 2x - 2 = 0
b) -2x2 + 8x - 4 = 0
d) a + 2a + 4 = 0
2
g) 2y2 + 19 = 12y
6.
x2
15
e)
= x+
2
2
c) -5x2 + 4x = -2
f)
1 2
40
m −m=
3
3
a)
An astronaut jumps from her spacecraft to the surface of the moon.
Her height, h in metres, above the ground after t seconds is modelled by
the function h = -0.8t2 + 2t + 8.
i)
Graph the function. What domain should you use?
i)
How long does it take her to reach her maximum height?
ii)
What is her maximum height?
iii)
How long does it take for her to reach the ground? Do you
use a quadratic function or a quadratic equation to find
this answer? Explain.
b)
A stone is dropped into a river from a cliff. Its height, h in metres,
above the water after t seconds is modelled by the function h = -4.9t2 +
82.
i)
How long does it take the stone to reach the water?
ii)
How long does it take to reach a height of 52m?
c)
A pitcher tosses a softball to a batter. Its path is modelled by the
function
h =-0.02d2 + 0.40d + 1.00, where d is the horizontal distance travelled
in metres and h is the height in metres.
i)
If the batter misses the ball completely, how far away from the
pitcher will it land?
ii)
If the batter hits the ball at a height of 1.00m, how far is the
batter from the pitcher?
d)
A department store sells shirts. The number sold, S, depends on the
price, p in dollars and is modelled by the function S = 0.05p2 - 6.50p +
249.68, provided $0.00 # p # $60.00. At what price(s) can the store
expect to sell 84 shirts?
e)
(Math 3205 only)
Exploits Valley Outfitters provides hunting and fishing guides for
Page -18-
people coming from outside the province. Last year, there were 1020
guests who each paid $180 per night. Management estimates that for
each $1.00 reduction in price there will be 5 extra customers.
i)
Algebraically obtain a function to model the expected
number of customers, C, at price p in dollars.
ii)
Write another function to model the total revenue, R in
dollars, as a function of p.
iii)
At what price would there be no guests, assuming p > 0?
iv)
What is the maximum revenue that can be obtained? At
what price does this occur?
7.
One root of each quadratic equation is given. Find the value of a, b or c as
indicated, then find the remaining root.
a)
c)
8.
8x2 - 6x + c = 0;
3x2 = -14x + c;
Solve:
a) x + 1 =
e)
10.
2x2 + bx + 6 = 0;
73z + 45 = az 2 ;
2
-5/7
b) -4, -8
6
x
b)
c)
1 3
,−
2 4
8
= x− 3
x−1
d)
10,− 10
e)
c)
4
= x+1
x− 2
d)
Solve:
4
3
+
=5
x x+2
8
7
d)
+
= −1
x +2 1− x
−2 5
− =2
x+3 x
8
6
e)
=
−3
x+3 x−1
b)
x+2
= x−4
x−2
4
6
+
= −7
x−2 x+1
5
9
f)
−2=
3x − 1
6x − 1
c)
Solve:
a)
2
2
,−
2
2
6−x
= x+ 5
x+4
a)
11.
b)
d)
Create a quadratic function for each given set of zeros. Make sure the
coefficients are integers.
a) 3, -5
9.
0
-2/3
3x
x
2x − 1
+
=
x−2 x+ 2 x+2
b)
5x + 2
2x
x
=
−
x+ 3 x+ 3 x−3
Page -19-
2x − 3 x + 2
3x
c)
−
=
x+2 x−1 x−1
3x2
x
x
d) 2
=
+
x −1 x+ 1 1− x
2
e) 2x − x = -5
x2 − 9 3 -x x +3
12.
a)
A crew rowed downstream from their campsite and back again,
travelling a total distance of 6km in 2 23 h . If the current was flowing
at a rate of 3km/h, what was the crew’s rowing rate in still water?
b)
Marian can cross-country ski 1 km/h faster than Larry. In a race, they
will both be timed to see how long they take to ski 4 km. Marian starts
1 minute later than Larry and overtakes him exactly at the finish line.
What is each skier’s average speed?
c)
Mr. I.M. Outofit finds that in clear weather he drives 10km/h faster
than if it is raining lightly and it takes 28 minutes less to travel from
Botwood to St. John’s, a distance of approximately 420km. What
average speed does he travel in light rain?
d)
At Pat’s Couch Potato Sports Store , a quantity of baseballs is ordered
for a cost of $600. If they had each cost $0.25 less, the store would
have received 10 more baseballs. How much is each ball?
Investigation 5
13.
Solve each equation in #3 and #5 using the quadratic formula.
14.
Find the roots of each of the following equations using the quadratic formula:
a)
1 2
x + 11x + 12 = 0
2
d) 5x2 + 20 = 0
15.
b) x2 +
e)
8x + 1 = 0
3x2 − 2x = − 3
c) 8m2 - 9m = 0
f)
2x 2 −
Find the x-intercepts of each of the following functions:
1 2
x − 4x − 10
2
a) y = (x + 3)2
b) f(x) = x2 - 8x - 9
c) y =
d) y - 3 = (x + 1)2
e) g(t) = c(x-3)2 + 18
f) y = 2(x - 4)2 + 1
Page -20-
8 = − 4x
Page -21-
16.
Find the vertex of each function algebraically without completing the square:
a) y = x2 + 10x + 4
4x2 + 8x
d) y = −
b) y = -3x2 - 6x + 1
1 2
x − 20x + 18
2
e) y = -4.9x2 + 9.8x + 14.7
g) y =
f) y = -3.1x2 + 12.4x
17.
c) y - 2 =
2x 2 + 2 2x + 2
Find the roots.
a)
1
1
−
=2
x+3 x− 3
b) −
d)
x+6 x
=
x
2
e)
1
= x+ 8
x
5x − 1 6x + 1
=
3x − 2
3x
c) x =
f)
− 3
3x − 2
7−x
=
5x
5x
7+x
Focus G
18.
Use a quadratic equation to solve each of the following algebraically:
a)
A rectangular ice rink is to be doubled in size by adding a strip of the
same width to one end and a side, as shown. What is the width of the
strips?
b)
Squares 4cm on a side are cut from the corners of a square piece of
sheet metal. The sides are then turned up to form an open box. If the
volume of the box is 1024 cm2, what is the original length of each side?
Page -22-
c)
Find two numbers whose sum is 17 and whose product is 72.
d)
A rectangle is 4cm longer than twice its width and its area is 96cm2.
What are its dimensions?
e)
The sum of the squares of two consecutive natural numbers is 145. What
are the integers?
f)
John found that the marks he could expect on a math test were related
to the number of hours he studied the night before the test by the
formula
m = -4h2 + 34h + 18, where m is the mark and h is the number of
hours studied.
i)
What number of hours should John study to obtain his maximum
mark?
ii)
What will be his mark if he does not study?
iii)
John also finds that if he has not studied consistently each night
and studies too long the night before, his mark will be lower
because he will be very tired. On his last test, his mark was zero.
How many hours did John study the night before the test?
g)
A rock garden is in the shape of a right triangle. The lengths of the legs
are respectively one metre and eight metres less than the hypotenuse.
What is the length of each side?
h)
The product of the slopes of two lines is -3. If the lines have slopes
represented by 3k+1 and -2k+5, respectively, what are the possible
value(s) of k?
i)
A rectangular flower garden measuring 16m by 2m is to have its area
increased by a factor of 225%. This will be accomplished by adding a
strip of the same width to one side and an end of the garden. What will
be the width of the strip?
Investigation 6
19.
What are the number and type of roots of each quadratic equation?
a) x2 + 2x + 9 = 0
b) x2 + 9x + 2 = 0
c) 3x2 - 8x - 5 = 0
d) 2x2 - 16x + 32 = 0
Page -23-
Page -24-
20.
Which function(s) have a corresponding quadratic equation with a negative
discriminant?
21.
Without calculating it, which of these functions has a positive discriminant? a
zero discriminant?
A) y = 2(x - 1)2 + 2
C) y = 2x2 + 8
B) -(y - 3) = (x + 1)2
D) y = (x - 31)2
22. (Math 3205 Only)
Find the values of k if the equation 10kx2 - 6kx + 9 = 0 has one real root.
23.
(Math 3205 only)
Explain why it is not possible for a quadratic function to have only one
complex zero.
24.
(Math 3205 only)
Explain why x2 + kx - 1 = 0 has real roots for all values of k.
25.
(Math 3205 Only)
a)
For what value of k does kx2 - 6x + 2 = 0 has equal real roots?
b)
For what values of k does x2 + kx - 8x + 9 = 0 have equal real roots?
c)
For what values of p does x2 + p x - x + ¼ = 0 have two unequal real
roots?
Investigation 7 (Math 3205 Only)
26.
Without solving, what are the sum and product of the roots of each equation?
a) x2 - 5x + 27 = 0
b) ½x2 - 4x = ¼
c) 4px2 - 4qx + 4r = 0
27.
a)
Write a quadratic equation whose roots have a product of -2 and a sum
of 3.
Page -25-
b)
Write a quadratic equation whose roots have a product of -27 and a sum
of 0.
c)
The roots of a quadratic equation have a sum of
4
7
and a product of -2.
Write the equation without fractional coefficients.
28.
Write a quadratic equation without fractional coefficients for each pair of
roots.
a) -3, 9
c) 3 ±
b) ¼, - ¾
3
29.
The equation 10x2 + c = 0 has one root equal to 3. What is the other root?
30.
The roots of ax2 + bx + c are negative reciprocals. What is the relationship
between a and c?
31.
The roots of 2kx2 + (3k+ 6)x -12 = 0 are p and q. If p + q = 0, what is the
value of k?
32.
The quadratic equation kx2 - (k - 6)x + 2 = 0 has roots m and n. For each
relationship, find the values of m and n:
a) m + n = 6
Answers
#1.
a) -1, -7 b) 0, 2
a) 2, -2 b) 2, 4
#2.
#3.
a) 2, -2 b) 2, 4
g) 3, -1/4
#5.
a) 1 ±
c) 0.5, 10
c) no real zeros
c)
d) 5.2, -1.2
c) -1, 2
h) -1/4, 1/7
d) -5/3
b) 2 ± /2
c)
1
mn = 12
e) -1, 2
e) 0, -3/7
f) -1/3, 4
2 ± 14
d) -1 ± i /3
e) 5, -3
5
f)
2 1
± i 2
3 3
g)
1
3± i 2
2
a) i) Domain is approximately {t* 0#t# 4.7, t0R}
ii) 1.25s iii) 9.25m
10
9
8
Height, h (m)
#6.
/3
b) mn = 8
7
6
5
4
3
2
1
1
iv)
2
3
Time, t (s)
4
5
4.7s; use a quadratic equation since we need to find an x-intercept and we can solve accurately using
the quadratic formula or completing the square.
Page -26-
b) i)
4.1s
ii) 2.5s
Page -27-
c) i)
22.25m
ii) 20m
d) i)
e)
i) C = 5(180 - x) or C = 900 - 5x
ii) R = x(900-5x) or -5x2 + 900x
$34.81
(the other root, $95.19, does not fit the domain
restriction)
#7.
iii) $384
a) c = 0, root = -3/4
iv) $184320; $192
b) b = -7; root = 3/2
c) c = -8; root = -4
d) a = -14; root = -9/2
#8
a) y = x2 + 2x - 15
b) y = x2 + 12x + 32
c) y = 8x2 + 2x -3
d) y = x2 - 10
e) y = 2x2 - 1
#9.
a) -3, 2
#10.
a) -1, -8/5
d) -6, 4
e) -5, 7/3
f) -1/12, 2/3
#11.
a)
#12.
a) 6 km/h
b) 5, -1
d) 1, 6
b) -5, -3/2
− 9 ± 97
4
#14.. a) -11 ±
c) -2, 3
b)
1± 7
2
e) -9, -2
c) -2, 11/7
c)
− 15 ± 217
4
b) Larry 15 km/h, Marian 16km/h
/97
b) -/2 ± 1
d) 0, -2/3
e)
− 4 ± 61
3
c) 90km/h
c) 0, 9/8
d) $4
d) ± 2i
e)
3 ±i 6
3
f) -/2 ±
2
#15. a) -3
b) 9, -1
c) 10, -2
d) no x-intercepts
#16. a) (-5, -21)
b) (-1, 4)
c) (-1, -2)
d) (-20, 218)
#17. a) ±/6
b) -/2 ± 1
c) 2, -3/2
d)
d) 16cm by 6cm
e) 8, 9
#18.
a) 10m
b) 24cm
h) 8/3, -1/2
#19.
c) 8, 9
− 1 ± 13
2
e) 15, -9
e) (-1, 0)
e)
4±
f) (2, 26.04)
3 ±2 3
3
f) i) 4.25 hours
f)
ii) 18
f)
b) Two different real roots
c) Two different real roots
d) One real root (two equal real)
iii) 9h
g) 5m, 12m, 13m
#20.
B) and D)
#22.
10
#23.
In the corresponding equation, to have a complex root the discriminant must be negative; therefore, this
produces a complex number preceded by a “±” when using the quadratic formula. In fact, the zeros must be
complex conjugates.
#24.
Discriminant is k2 + 4; if k is negative or positive, k2 is positive, so adding 4 is still positive. If k is 0, then the
discriminant is 4, which is positive.
#25. a) 9/2
#21. positive: B) Zero: D
b) 14 or 2
c) x < 0, x > 2
#26. a) sum = 5, prod. = 27
b) sum = 8, prod. = -½
#27. a) x2 -3x - 2 = 0
b) x2 - 27 = 0
#28. a) x2 - 6x - 27 = 0
b) 16x2 + 8x - 9 = 0
#29.
-3
#30.
a = -c
#31.
c) sum =
q
, prod. =
p
r
p
c) 7x2 - 4x - 14 = 0
c) x2 - 6x + 6 = 0
-2
#32. a) -6/5
Page -28-
b) 1/4
c) 24
g) (-1, 0)
− 5 ± 53
2
i) 20m
a) No real roots (Two unequal complex)
2
2