ALGEBRA UNIT 11-SOLVING QUADRATIC EQUATIONS
SOLVING QUADRATICS BY FACTORING (DAY 1)
Quadratic Equation (Standard Form):
HOW TO SOLVE QUADRATIC EQUATIONS:
Step 1:
Write equation in Standard Form.
Step 2:
Factor the quadratic equation.
Step 3:
After the problem has been factored we will complete a step called the “T”
chart. Create a T separating the two ( ).
1.
3.
Step 4:
Once ( ) are separated, set each ( ) = to 0 and solve for the variable.
Step 5:
Check each of the roots in the ORIGINAL quadratic equation.
Find the roots: r 2  12r  35  0
Find the zeroes:
x 2  5x  6  0
Solve for y: y 2  11y  24  0
2.
4.
1
Solve for y:
y 2  3y  28
x 2  x  30
6.
Find the roots:
7.
A plot of land for sale has a width of x ft., and a length that is 8ft less than its width. A
farmer will only purchase the land if it measures 240 square feet. What value of x will
cause the farmer to purchase the land?
8.
The length of a rectangle is 5 inches more than twice a number. The width is 4 inches
less than the same number. If the area of the rectangle is 15, find the number
2
Find the zeros:
5w 2  35
5.
QUADRATIC WORD PROBLEM (DAY 2)
CONSECUTIVE INTEGERS/GEOMETRIC PROBLEMS
Review of Consecutive Integers “Let Statements”:
 Consecutive Integers:

Consecutive Even Integers:

Consecutive Odd Integers:
1.
Find two consecutive odd integers whose product is 99.
2.
A certain number added to its square is 30. Find the number.
3.
The square of a number exceeds the number by 72. Find the number.
4.
Find two consecutive positive integers such that the square of the first decreased by
17 equals 4 times the second.
3
5.
The ages of three family children can be expressed as consecutive integers. The
square of the age of the youngest child is 4 more than eight times the age of the
oldest child. Find the ages of the three children.
6.
The altitude of a triangle is 5 less than its base. The area of the triangle is 42 square
inches. Find its base and altitude
7.
The length of a rectangle exceeds its width by 4 inches. Find the dimensions of the
rectangle if its area is 96 square inches.
8.
If the measure of one side of a square is increased by 2 centimeters and the measure
of the adjacent side is decreased by 2 centimeters, the area of the resulting rectangle
is 32 square centimeters. Find the measure of one side of the square (the original
figure).
4
SIMPLIFYING RADICALS (DAY 3)
What is a perfect square? ___________________________________________________
What happens when you take the square root (
) of a perfect square (VIPS)?
When solving quadratics, the answer is not always a perfect square when you take the
a number.
In order to find the answer you will need to SIMPLIFY the
of non- perfect squares.
PROCEDURE TO SIMPLIFY RADICALS (non-perfect squares):
1.
List perfect squares from 1 to 169 out. (VIPS) (create a list)
2.
Determine where the number that you are simplifying would fall in this list.
3.
Work UP the list to #1 to locate the largest perfect square that goes into your
number.
4.
Once found, break up your original
(house) by placing the largest
perfect square number found under the first
under the
5.
second
sign and place its divisor
sign.
Circle the perfect square. – IT IS SO PERFECT THAT …
Simplify the following:
1)
18
2)
- 4 98
3) -
4)
1
12
2
5)
2 50
6)
5
of
3
48
4
27
VIPS LIST
7)
9 99
10)

4
25
8)
3 32
9)
2
75
5
11)
 0.16
12)
4 500
Pythagorean Theorem Formula: _________________________________________
13)
Given a right triangle with leg of 8 m and a hypotenuse of 12m, determine
the length of the other leg in simplest radical form.
14)
Given a right triangle with a hypotenuse of 8 2 and a leg of 10. Find the length of the
missing leg in simplest radical form.
15.
Given a right triangle with a leg of 8cm and a leg of 6cm. Find the length of the
hypotenuse in simplest radical form.
6
SIMPLIFYING RADICALS CONT…(DAY 4)
VIPS LIST
1)
5 64
2)
 2 48
3)
1
72
2
4)
2 x 128
5)
 5 176
6)

7)
3 147
8)
 2 500
9)
5a2 24
7
1 4
d 50
2
10)
The length of the hypotenuse of a right triangle is 13cm. One leg is 7cm, find the
length of the other leg in simplest radical form.
11)
A woman casts a shadow of 11ft. The measure from the top of her head to the tip of
the shadow is 12.5ft. How tall is the woman to the nearest foot?
12)
Given a right triangle with a hyptoneuse of 9 2 and a leg of 8, find the length of the
missing leg in simplest radical form.
8
SOLVING QUADRATICS THAT CAN’T BE FACTORED
OPTION 1: COMPLETING THE SQUARE (DAY 5)
1) What is a quadratic equation?
2) What are the zeros (roots) of a quadratic equation? (algebraically and graphically)
3) What are some methods you know of to find the roots of a quadratic equation?
4) Factor:
x 2  8 x  16
5) x 2  8 x  16 is an example of a perfect square trinomial. Explain why.
Procedure for Completing the Square
1. Rearrange the equation:
 Get terms with variables on the left hand side.
 Get c# by itself on the right hand side
2. If a# > 1 then divide through by a#.
3.

Identify the b#
 (half it - square it - add it to both sides)
(This will form a perfect square trinomial)
4.

Write Expression as a perfect square trinomial

Simplify the # on the other side


Square root both sides
Put a  on the right in front of term
5.
6. Solve for x to find the roots.
9
2x2 – 12x + 14 = 0
Solve each quadratic equation by completing the square; express each root in simplest
radical form.
1)
x 2  2x  12
2) 3x 2  6 x  24  0
3)
1 2
r  6r  2
2
4)
10
2y 2  3y  5  4
COMPLETING THE SQUARE CONT… (DAY 6)
Recall basic steps:
1)
Solve 2x 2  12x  4  0 by completing the square; express the result in simplest form.
2)
Brian correctly used a method of completing the square to solve the equation
x 2 + 7x – 11 = 0. Brian’s first step was to rewrite the equation as x 2 + 7x = 11. He then
added a number to both sides of the equation. Which number did he add?
1)
3)
2)
49
4
3)
49
2
4) 49
If x2 + 2 = 6x is solved by completing the square, an intermediate step would be
(1)
(2)
(3)
(4)
4)
7
2
(x + 3)2 = 7
(x – 3)2 = 7
(x – 3)2 = 11
(x – 6)2 = 34
The senior class at Bay High School buys jerseys to wear to the football games. The
cost of the jerseys can be modeled by the equation C=0 .1x 2  2.4x  25 , where C is
the amount it costs to buy x jerseys. How many jerseys can they purchase for $430?
11
5)
Collin is building a deck on the back of his house. He has enough lumber for the deck
to be 144 square feet. The length should be 10 feet more than its width. What should
the dimensions of the deck be?
6)
The product of two consecutive negative odd integers is 483. Find the integers.
7)
Solve p2  3p  8 by completing the square.
12
SOLVING QUADRATICS THAT CAN’T BE FACTORED
OPTION 2: QUADRATIC FORMULA (DAY 7)
Recap:
Solve 7p2  12p  4  0 by completing the square
Completing the square and factoring are not always the best method to use when solving a
quadratic as illustrated above.
To DERIVE a new method to solve a quadratic we go back to the beginning.
DERIVE QUADRATIC FORMULA USING
COMPLETING THE SQUARE
ax 2  bx  c  0
1. Rearrange the equation:
 Get terms with “x” on the left hand side.
 Get c# (variable) by itself on the right hand side
2. Divide through by a.
3.
Identify the b#
(half it - square it - add it to both sides)
(Since there are no #’s to half—literally show multiplying by
Square this “expression part”—add it to both sides
1
)
2
4.
Write Expression as a perfect square trinomial
 Simplify the other side (adding fractions need common
denominators)--Answer should be in Standard Form.
5.
Square root both sides
Put a  on the right in front of term
 (Fractions already have Common Denominator) – just
write final expression.
13
6. NOW you have a new formula called the
Quadratic Formula
(you can remember this formula through a song)
Steps for using the Quadratic Formula: (Use when quad equation = 0)

Get equation equal to ZERO!!!

Put Equation in standard form: __________________________

Identify the a, b, and c #’s.

Plug into the formula and simplify.
To remember formula sing/hum the phase below to the “pop goes the weasel song”
“x =’s negative b, plus or minus the square root of b2 minus 4 a c, all over 2 a”
******WRITE THE FORMULA DOWN AS YOU SING THE SONG******
Solve the following using the quadratic formula, answers should be in simplest radical form
when possible.
1)
Solve for x:
x2 – 12x = -20
2)
14
Solve for x:
2x2 + 9x = 18
Quadratic Formula:
3) Solve for x:
x2 – 5x – 3 = 0
5) Solve for x:
5x2 – 2 = 3x
4) Solve for x:
15
2x2 – 4x = 1
QUADRATIC FORMULA – CONTINUED (DAY 8)
Quadratic Formula:
What is the quadratic formula used for?
Give some reason(s) to use the quad formula instead of completing the square method.
Solve the following using the quadratic formula. Answer should be in simplest radical form
when possible.
1)
Solve for x:
-3x 2 = 8x – 12
2) Solve for x:
16
3x2 + 5x – 12 = 0
3)
The Demon Drop at Cedar Point in Ohio takes riders to the top of a tower and drops
them 60 feet. A function that approximates this ride is h= -16t 2 + 64t + 60, where h is
the height in feet and t is the time in seconds. About how many seconds does it take
for riders to drop 60 feet?
4)
The percent of U.S. households with high speed Internet h can be estimated by
h = -0.2n 2 + 7.2n + 1.5, where n is the number of years since 1990. Use the Quadratic
Formula to determine when 20% of the population will have high speed Internet.
5)
Solve the following quadratic equation, x 2 = -7x – 5, selecting a method that we
learned, and state why you chose that method.
17
METHODS OF FINDING ROOTS (ZEROS) OF QUADRATIC EQUATIONS (DAY 9)
METHOD
REASON TO USE THIS METHOD
Find the roots by FACTORING:
1) x2 – 12x = -27
2) 2x2 + 3 = -7x
18
QUICK PROCDURE
Find the roots by FACTORING:
4) 5 x 2  15x
3) x2 – 64 = 0
Solve the equation using the QUADRATIC FORMULA, leave answers in simplest radical form.
5)
x2 – 5x – 3 = 2
6)
x 2  3x  4  50
Find the roots by COMPLETING THE SQUARE, leave answers in simplest radical form.
9)
3x2 - 2x – 4 = 0
10)
19
2 x 2  5x  2  0
20