Download 1 Study Guide #4: Quadratic Functions and Complex Numbers

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Study Guide #4: Quadratic Functions and Complex Numbers
Section 1: Complex Numbers
The number i (which is not a real number) is called the imaginary unit.
i  1
If x is a positive number, then:  x  i x
Always remove the i first before doing any calculations.
Ex: 7  1 7  i 7
We frequently use expressions which involve i 2.
Since i  1 , then i 2 

1

2
 1, therefore, i 2  1
Examples: Simplify:
a) 24
b) 4 75
c) 2
d)
e)
f)
g)
25  4
4 14
32 7
6  8
3
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50  2
2
Powers of i
Do you notice a pattern?
i0  1
i4  1
i8  1
i1  i
i5  i
i9  i
i 2  1 i 6  1 i10  1
i 3  i i 7  i
i11  i
The powers of i repeat their values in cycles of 4. Any power of i has the same value of i0, i1,
i2, i3.
To find the value of any power of i, divide the exponents by 4 and look at the remainder.
The remainder is the power of i that the value is equivalent to.
When dividing on the calculator, remainders are: .0=R0
.25=R1
.5=R2
.75=R3
Examples:
a) i 34
b) i17
Can we do this on the calculator?
MATHNUM-3:iPart(
c) i 40
d) i 27
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Complex number is a number that can be written in the form a + bi where a and b are real
numbers and i  1 .
In a + bi :
If a = 0, then 0 + bi = bi (pure imaginary)
If b = 0, then a + 0i = a (real number)
Examples:
3 + 4i
7 – 5i
6i
11 – 8i
Graphing Complex Numbers:
To graph a complex number of the form a + bi, we must change our coordinate plane into a
complex number plane.
Let the horizontal axis represent the real number line and the vertical axis represent the
pure imaginary number line.
A complex number can be represented as a point or
by a vector (shows magnitude and direction with an
arrow). The tip of the arrow must be on the point.
The endpoint is located at the origin.
Examples: Graph:
a) 3 + 4i
b) 5 – 2i
c) -4 – i
d) -2 + 3i
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Section 2: Operations with Complex Numbers
Two complex numbers are equal if their real parts are equal and their imaginary parts are
equal.
a + bi = c + di means a = c and b = d
Example: Solve for x and y if
3x + 4yi = -12 + 20i
3x = - 12
and
4y = 20
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3
4
4
x = -4
y=5
Additive inverse: Remember when the inverse is added to a number the result is the
identity or zero. Therefore the additive inverse of a + bi is –a – bi.
Examples: Find the additive inverse:
a) 2 + 4i
b) 6 – 3i
c) -5 + 2i
d) -1 – 7i
Adding Complex Numbers:
(3 + 4i) + (5 + 6i )
Add the real parts 3 + 5 = 8; Add the imaginary part 4i + 6i = 10i
So (3 + 4i) + (5 + 6i) = 8 + 10i
Subtracting Complex Numbers:
(4 + 3i) – (5 + 2i)
means (4 + 3i) + (-5 – 2i) = -1 + i
Examples:
a) (9  10i)  (7  6i)
b) (8  i )  (1  4i )
c) (2  5i)  (6  3i)
d) (5  2i)  (8  7i)
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Multiplying Complex Numbers:
(3 – 4i)(5 + 2i)
Use FOIL
15 + 6i – 20i – 8i2
15 – 14i -8(-1)
15 – 14i + 8
23 – 14i
Examples:
a) (2  5i)(4  6i)
b) (1  3i)(8  2i)
c) (3  5i)(2  3i)
d) (2  3i) 2
e) (4  5i )(4  5i )
f) (3  2i )(3  2i )
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Conjugate
The conjugate of a + bi is a – bi
Examples: Find the conjugate of each:
a) 6 – 7i
b) -2 – 5i
c) -3 + i
Division of Complex Numbers:
To rationalize a fraction whose denominator is in the form a + bi, we must use its conjugate.
Example:
a)
2i
3i
b)
3  5i
1  2i
c)
6i
3i
d) Express the multiplicative inverse of -3  2i .
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Section 3: Real Roots of a Quadratic Equation
The real roots of a polynomial function are the x-coordinates of the points where the graph
crosses the x-axis.
Graph: x2 – 5
Using the calculator we see that the roots appear to be ______________________.
Now, solve for the roots algebraically:
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However some equations cannot be solved as easily. We can use a process called completing
the square:
(1) Be sure that the coefficient of the highest power is one. If it is not, divide each term by
that value to create a leading coefficient of one.
Ex: x2 + 8x – 4 = 0
(2) Move the constant term to the right hand side.
x2 + 8x – 4 = 0
+ 4 +4
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x + 8x = 4
(3) Prepare to add the needed value to create the perfect square trinomial. Be sure to
balance the equation. The boxes may help you remember to balance.
x2 + 8x + _____ = 4 + _____
(4) To find the needed value for the perfect square trinomial, take half of the coefficient
of the middle term, square it, and add that value to both sides of the equation.
x2 + 8x + 16 = 4 + 16
(5) Factor the perfect square trinomial.
Hint: What it was when you took ½.
(x + 4)2 = 20
(6) Take the square root of each side and solve. Remember to consider both plus and minus
results.
x
 4 
2
x4
20
4 5
x4 2 5
4 4
x  4  2 5
Examples: Solve by completing the square. Express in simplest radical form.
a) x2 – 10x + 7 = 0
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b) x2 + 8x – 12 = 0
c) 3x2 – 6x – 1 = 0
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Section 4: Quadratic Formula
Quadratic Equation: ax2 + bx + c = 0
Quadratic Formula: x 
b  b2  4ac
2a
The Quadratic Formula must be MEMORIZED.
Examples: Solve using the quadratic formula.
a) x2 – 10x + 7 = 0
b) 2x2 = 7 – 3x
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Quadratic Formula – Word Problems:
c) The length of a rectangle is 8cm more than its width. If the area is 12cm2 find the
dimensions in simplest radical form.
d) Barb pulled the plug in her bathtub and it started to drain. The amount of water in the
bathtub as it drains is represented by the equation
, where L represents the
number of liters of water in the bathtub and t represents the amount of time, in minutes,
since the plug was pulled. How many liters of water were in the bathtub when Barb pulled
the plug? Show your reasoning. Determine, to the nearest tenth of a minute, the amount of
time it takes for all the water in the bathtub to drain.
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Section 5: Discriminant
The discriminant is the name given to the expression that appears under the square root
(radical) sign in the quadratic formula. Discriminant of the quadratic equation: b2 – 4ac.
The discriminant tells you about the "nature" of the roots of a quadratic equation given that
a, b and c are rational numbers. It quickly tells you the number of real roots, or in other
words, the number of x-intercepts, associated with a quadratic equation.
Ex: Find the discriminant of x2 + 7 = 5x.
Value of the
Graphs Showing x-intercepts
DISCRIMINANT
y = ax2 + bx + c
Nature of the Roots
Positive & Perfect Square:
Real, Rational, Unequal
POSITIVE
There are 2 real roots.
Positive & Not a Perfect
Square:
Real, Irrational, Unequal
Real, Rational, Equal
ZERO
There is 1 real root.
Imaginary
NEGATIVE
There are NO real roots.
There are 2 imaginary roots.
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Examples:
1. Suppose the discriminant is the given number. (a) Describe the nature of the roots. (b)
Determine the number of x-intercepts of the graph.
(1) 15
a) ___________________________
b) ___________________________
(2) 0
a) ___________________________
b) ___________________________
(3) -9
a) ___________________________
b) ___________________________
2. Describe the nature of the roots of 3x2 – x – 5 = 0.
3. Determine the value(s) of k for which the given equation has the given type of roots:
(a) x2 + 2x + k = 0 ; 2 real roots
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(b) 3x2 + 4x + k = 0 ; 2 imaginary roots
(c) 2x2 – 8x + k = 0 ; 1 equal root
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Section 6: Complex Roots of a Quadratic Equation
We can use the quadratic formula to solve equations with complex roots.
Examples: Solve and express the roots in simplest a + bi form.
(a) x2 + 17 = 8x
(b) 9x2 = 6x – 2
_________________________________________________________________
(c) 2x2 = -3(2x + 3)
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Section 7: Sum and Product of the Roots of a Quadratic
Quadratic equation: ax2 + bx + c = 0.
Must memorize these formulas:
Sum of roots: r1  r2 
b
a
Product of roots: r1r2 
Examples:
1. Find the sum and product of the roots:
(a) x2 + 5x – 3 = 0
(b) 3x2 – x + 5 = 0
(c) -2x2 = 3x + 4
c
a
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2. One of the roots is given. Find the other root.
(a) x2 – 5x + c = 0 ; r1 = 2
(b) 2x2 – 12x + c = 0 ; r1 = 3  2
3. Find the value of k if one root is given.
(a) 2x2 – 6x + k = 0 ; r = -1
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(b) x2 + kx – 7 = 0 ; r = -1
4. Write a quadratic equation in standard form with the given roots:
(a) 4, - 6
(b) -7, -3
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Complex roots occur in conjugate pairs. If a + bi is one root, a – bi is the other root.
(c) 4 + 2i , 4 – 2i
(d) 3i, -3i
(e) 3  5 , 3  5
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Section 8: Solving Higher Degree Polynomial Equations
We can find the rational roots of a polynomial function by FACTORING. We can also show
that these are exact roots by substituting these roots in the equation to show that they
satisfy the equation.
REMEMBER: Types of Factoring:
(1) Greatest Common Factor
(2) Difference of 2 Perfect Squares
(3) Trinomial
(4) Master Factoring
(5) Group Factoring
Examples:
1. Find all roots of the given equation.
(a) x3 + x2 – 4x – 4 = 0
(c) x3 + 2x2 + 4x + 8 = 0
(b) x4 – 10x2 + 9 = 0
(d) x4 – 9x3 – 10x2 = 0
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2. Find f(a) for each function. Is “a” a root?
(a) f(x) = x4 + 8x3 + 22x2 + 24x + 9 and a = -1
(b) f(x) = x3 – 2x2 + 9x – 18 and a = 3i.
(c) f(x) = x3 – x2 – 5x – 3 and a = 1 + i.
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Section 9: Solutions of Systems of Equations and Inequalities
A quadratic-linear system consists of a quadratic equation (parabola) and a linear function
(straight line). The system may have:
(a) 2 solutions ; 2 points of intersection
(b) 1 solution ; 1 point of intersection
(c) No solution ; 0 points of intersection
We can use either a graphic or algebraic method to solve the system of equations.
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Examples:
1. Solve y = x2 – x – 6 and y = 2x – 2
Graphically:
Algebraically:
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Graphing a quadratic inequality in 2 variables:
(1) Graph the corresponding quadratic function
(2) Decide whether to use a solid (< or >) or broken (< or >) curve.
(3) Test a point (usually (0,0)) in the inequality and shade accordingly.
Examples:
1. Graph y < -2x2 + 4x + 7
2. Graph y + 6 < x2 – 4x
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Review of Solving Quadratic Inequalities:
Examples:
1. Write the solution set of each inequality:
(a) x2 + 5x – 6 > 0
(b) x2 – 3x +2 > 0
____________________________________________________________________
Regents Word Problems:
1. The height of a projectile is modeled by the equation
, where x is time, in
seconds, and y is height, in feet. During what interval of time, to the nearest tenth of a second, is
the projectile at least 125 feet above ground?
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2. The profit a coat manufacturer makes each day is modeled by the equation
where P is the profit and x is the price for each coat sold. For what values of x does the company
make a profit?
3. A small rocket is launched from a height of 72 feet. The height of the rocket in feet, h, is
represented by the equation
, where
, in seconds. Graph this equation on
the accompanying grid. Use your graph to determine the number of seconds that the rocket will
remain at or above 100 feet from the ground. [Only a graphic solution can receive full credit.]
,