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MATH10 ALGEBRA
QUADRATIC EQUATIONS
Quadratic Equations (Algebra and Trigonometry, Young 2nd Edition, page 113-135)
GENERAL OBJECTIVE
At the end of the chapter the students are expected to:
Week 3 Day 3
• Solve quadratic equations using different methods,• Solve equations in quadratic form,• Solve equations leading to quadratic equation, and • Solve real-world problems that involve quadratic equation.
TODAY’S OBJECTIVE
• To distinguish between pure quadratic equation and complete quadratic equation,• To determine the number and type of solutions or roots of a
quadratic equation based on the discriminant,• To define complex numbers, and• To solve quadratic equations by factoring, square root method,
completing the square and quadratic formula.
At the end of the lesson the students are expected to:
Week 3 Day 3
A quadratic equation in x is an equation that can be written in the standard form where a, b, and c are real numbers and a 0 .
DEFINITION
QUADRATIC EQUATION
Example:
Week 3 Day 3
0cbxax 2
03x3x5 2
0x2 2
0x7x2 2
a represents the numerical coefficient of x2 , b represents the numerical coefficient of x, and c represents the constant numerical term.
050x2 2
• Pure Quadratic Equation
If b=0, then the quadratic equation is termed a "pure" quadratic equation.
Example: 3x2 +6=0
• Complete Quadratic Equation
If the equation contains both an x and x2 term, then it is a "complete" quadratic equation.
The numerical coefficient c may or may not be zero in a complete quadratic equation.
Example: x2 +5x+6=0 and 2x2 - 5x = 0
Week 3 Day 3
The term inside the radical, b2 -4ac, is called the discriminant.The discriminant gives important information about the corresponding solutions or roots of where a, b, and c are real numbers and a 0 .
DEFINITION
DISCRIMINANT OF A QUADRATIC EQUATION
Week 3 Day 3
0cbxax 2
b2 -4ac Solutions or Roots
PositiveZero
Negative
Two distinct real rootsOne real root (a double or repeated root)Two complex roots(complex conjugates)
036 .5
04129 .4
09124 .3
052 .2
054 .1
2
2
2
2
2
xx
xx
xx
xx
xx
Determine the nature of roots of the following quadratic equation.
Week 3 Day 3EXAMPLE
A complex number is an expression of the form where a and b are real numbers and a is the real part and b is the imaginary part .
bia 1i 1i 2
Real Part Imaginary Part
i43
i3
2
2
1 2
1
3
2
3 4
i6 0 6
-7 -7 0
DEFINITIONCOMPLEX NUMBER
EXAMPLE
Week 3 Day 3
SOLVING QUADRATIC EQUATIONS
There are four algebraic methods of solving quadratic equation in one variable, namely:
• solution by factoring
• solution by square root method
• solution by completing the square
• solution by quadratic formula
Week 3 Day 3
SOLVING QUADRATIC EQUATIONS BY FACTORING
STEPS:1. Write the equation in standard form ax2 + bx + c = 0.2. Factor the left side completely.3. Apply the Zero Product Property to find the solution set.
The Factoring Method applies the Zero Product Property which states that if the product of two or more factors equals zero, then at least one of the factors equals zero. Thus if B·C=0, then B=0 or C=0 or both.
Week 3 Day 3
035x12x 114.pp
ex.1.3.1 Classroom .1 2
26t103t 115.pp
1.3.2 ex. Classroom .2 2
y125y 95.pp
1.3.3 ex. Classroom .3 2
4p129p 124.pp
#13 .4 2
Solve the following equations.EXAMPLE
025v16 124 page
16# .5 2
3x4x2
.8
13)1x3)(1x4( .73)5x2(x 6.
Week 3 Day 3
SOLVING QUADRATIC EQUATIONS BY SQUARE ROOT METHOD
The Square Root Property states that if an expression squared is equal to a constant , then the expression is equal to the positive or negative square root of the constant. . , xif Thus, 2 PxthenP
solutions imaginary two exactly has Px equation the ,0P If .3
zero of root double a has Px equation the ,0P If .2
Px and Px ;solutions real distinct 2 real has
Px equation the ,number real a is 0P If .2
1) to equal tcoefficien ( first isolated be must squared variable The .1:NOTE
2
2
2
Week 3 Day 3
0322x 116.
ex.1.3.4 Classroom .1 2
pp
0105a 117.pp
1.3.5 ex. Classroom .2 2
253)-(x 117.pp
1.3.6 ex. Classroom .3 2
161)-(4x 124.pp
#30 .4 2
Solve the following equations.EXAMPLE
025v16 124 page
16# .5 2
112 page 10 and 9 numbers 5.4 exercise Sabino and Marquez Exconde, by Algebra College from
3m5m2 .7
7y4y2 .6
22
22
Week 3 Day 3
SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE
cbxx
form following the in equation quadratic the Express 1.2
222
2b
c2b
bxx
sides. both to square the add then result, the square and 2 by Divide 2.
b
STEPS:
Week 3 Day 3
SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE
22
2b
c2b
x
square perfect a as equation theof side left the Write 3.
method. root square the using Solve 4.
Week 3 Day 3
03x8x 118.pp
#7 Example .1 2
013x123x 119.pp
#2 Example .2 2
41
x22x
124.pp
#55 .3
2
065
3t2
3t
124.pp
#56 .4
2
Solve the following equations.EXAMPLE
x12x 124 page
#41 .6
x52
x 124 page
#41 .5
.expression eachof square the complete to added be should number What
2
2
Week 3 Day 3
SOLVING QUADRATIC EQUATIONS BY QUADRATIC FORMULA
THE QUADRATIC FORMULA
The roots of the quadratic equation ax2 + bx + c = 0, where a, b, and c are constants and a 0 are given by:
a
acbbx
2
42
c. b, a, parameters the identify to order in 0cbxax
form standard in be must equation quadratic The :Note2
Week 3 Day 3
DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE
Consider the most general quadratic equation: 0cbxax2
Solve by completing the square: WORDS MATH
.
ac
ab
2
a2b
0ac
xab
x2
ac
xab
x2
ac
a2b
a2b
xab
x22
2
2
22
a4ac4b
a2b
x
1. Divide the equation by the leading coefficient a.
2. Subtract from both sides.
3. Subtract half of and add
the result to both sides.
4. Write the left side of the equation as a perfect square and the right side as a single fraction.
DERIVATION OF QUADRATIC FORMULA BY COMPLETING THE SQUARE
WORDS MATH
a2b
2
2
a4ac4b
a2b
x
a2ac4b
a2b
x2
a2ac4bb
x2
5. Solve using the square root method.
6. Subtract from both sides
and simplify the radical.
7. Write as a single fraction.
8. We have derived the quadratic formula.
x22x 121.pp
Turn Your .1 2 0144x
121.
#11 Example .2 2 x
pp
Solve the following equations using the quadratic formula.
EXAMPLE Week 3 Day 3
SUMMARY
The four methods for solving quadratic equations are:
1. factoring
2. square root method
3. completing the square
4. quadratic formula
Factoring and the square root method are the quickest and easiest but cannot always be used.
Quadratic formula and completing the square work for all quadratic equations and can yield three types of solutions:
1. two distinct real roots2.one real root (repeated)3.or two complex roots (conjugates of each other)
Week 3 Day 3
EQUATIONS IN QUADRATIC FORM(OTHER TYPES)
Week 4 Day 1
12.01.0 x125 page
#93 .4
3-xx
3--
x
3
3
2-x4
125 page
#91 .3
7x
12 x
125 page
#89 .2
5
1
3
4
3
2
125 page
#87.1
method.any usingequation quadraticeach Solve
2
2
x
x
tt
CLASSWORKWeek 4 Day 1
TODAY’S OBJECTIVE
• To find the sum and product of roots of a quadratic equation.• To find the quadratic equation given the roots.• To transform a difficult equation into a simpler linear or quadratic
equation,• To recognize the need to check solutions when the transformation
process may produce extraneous solutions,• To solve radical equations.
At the end of the lesson the students are expected to:
Week 4 Day 1
RECALL
The four methods for solving quadratic equations are:
1. factoring
2. square root method
3. completing the square
4. quadratic formula
Factoring and the square root method are the quickest and easiest but cannot always be used.
Quadratic formula and completing the square work for all quadratic equations and can yield three types of solutions:
1. two distinct real roots2.one real root (repeated)3.or two complex roots (conjugates of each other)
Week 4 Day 1
SUM AND PRODUCT OF ROOTS
Recall from the quadratic formula that when
a2ac4bb
x 0cbxax2
2
a2
ac4bbs
a2
ac4bbr
s and r be roots the Let2
2
Week 4 Day 1
SUM OF ROOTS
a
ba2
b2sr
a2
ac4bbac4bb
a2
ac4bb
a2
ac4bb
22
22
sr
s r
Sum of roots = r + s
Week 4 Day 1
PRODUCT OF ROOTS
a
c s)((r)
a4
ac4bb )s)(r(
a4
ac4b)b(
a2
ac4bb*
a2
ac4bb ) s (r)(
2
22
2
222
22
Product of roots = (r) (s)
Week 4 Day 1
EXAMPLE
Determine the value of k that satisfies the given condition
signsopposite
withbut equaly numericall roots the 0; 6- x5-k x1-2k 4.
0. is roots the of one 0; 6 5k k10x-x12k 3.
1.- is roots of product 0;1-k2x2)x(3k 2.
20. is roots of sum0;45xxk 1.
2
22
2
2
Week 4 Day 1
FINDING THE QUADRATIC EQUATION GIVEN THE ROOTS
0sxr-x
is equation quadratic the s,and r be roots the Let
i21 and i21 3
27 and 25 2
2
1 and
4
3 1
.
.
.
Example: Find the quadratic equations with the given roots.
Week 4 Day 1
RADICAL EQUATIONS
Radical Equations are equations in which the variable is
inside a radical (that is square root, cube root, or higher
root).
62x72x ,x32x ,23x
Week 4 Day 1
RADICAL EQUATIONS
Steps in solving radical equations:
1. Isolate the term with a radical on one side.
2. Raise both (entire)sides of the equation to the power that will eliminate this radical and simplify the equation.
3. If a radical remains, repeat steps 1 and 2.
4. Solve the resulting linear or quadratic equation.
5. Check the solutions and eliminate any extraneous solutions.
Note: When both sides of the equations are squared extraneous solutions can arise , thus checking is part of the solution.
Week 4 Day 1
23-x 128.pp
1 Example .1
3x-2 .b
3x-2 a.
128.pp1.4.1 ex. Classroom
.2
3x62x 128.ppTurn Your
.3
62x72x 129.pp
3 Example .4
Solve the following equations.EXAMPLE
2x15x 133 page
28# .5
xx2 133 page
29# .6 3x1x82x
133 page24#
.7 2
Week 4 Day 1
made? wasmistake What incorrect. is This
5 t
153t
16 13t :Solution
413t equation theSolve
made. is that mistake eExplain th 134
83# .8
page
CATCH THE MISTAKE
Week 4 Day 1
SUMMARY
a2
ac4bbs
a2
ac4bbr
s and r be roots the Let2
2
a
b sr
:roots of Sum
a
c s)((r)
:roots ofProduct
Week 4 Day 1
.
Steps in solving radical equations:1. Isolate the term with a radical on one side.2. Raise both (entire)sides of the equation to the power that
will eliminate this radical and simplify the equation.3. If a radical remains, repeat steps 1 and 2.4. Solve the resulting linear or quadratic equation.5. Check the solutions and eliminate any extraneous solutions.
TODAY’S OBJECTIVE
• To solve equations that are quadratic in form,• To realize that not all polynomial equations are factorable.• To solve equations that are factorable.
At the end of the lesson the students are expected to:
Week 4 Day 2
EQUATIONS QUADRATIC IN FORM: u-SUBSTITUTION
Equations that are higher order or that have fractional powers often can be transformed into quadratic equation by introducing a u-substitution, thus the equation is in quadratic form.
Week 4 Day 2
Original Equation Substitution New Equation
Example:
04x3x 24
01t2t 31
32
2xu
31
tu
04u3u2
01u2u2
Steps in solving equations quadratic in form:
1. Identify the substitution.
2. Transform the equation into a quadratic equation.
3. Apply the substitution to rewrite the solution in terms the original variable.
4. Solve the resulting equation.
5. Check the solution in the original equation.
Week 4 Day 2EQUATIONS QUADRATIC IN FORM: u-SUBSTITUTION
012xx 131.pp
4 Example .1 12-
091x21012x 131.pp
1.4.4 ex. Classroom .2 12
Solve the following equations.EXAMPLE Week 4 Day 2
016z8z 131.pp
1.4.5 ex. Classroom .3 5
152
0xx2x 132.pp
1.4.5 ex. Classroom .4 3
235
38
Week 4 Day 2FACTORABLE EQUATIONS
EQUATIONS WITH RATIONAL EXPONENTS BY FACTORING
0x4x3x 132 page
6# Example 31
34
37
POLYNOMIAL EQUATION USING FACTORING BY GROUPING
04x2x126x 132 page
1.4.7 Ex. Classroom 23 03v403vv 133 page
66# 23
0y6y5y 132 page
71# 21
21
23
SUMMARY
Radical equations, equations quadratic in form, and factorable equations can often be solved by transforming them into simpler linear or quadratic equations. Radical Equations: Isolate the term containing a radical and raise it to the appropriate power that will eliminate the radical. If there
is more than one radical, it does not matter which radical is isolated first. Raising radical equations to powers may cause extraneous solutions, so check each solutions.
Equations quadratic in form: Identify the u-substitution that transforms the equation into a quadratic equation. Solve the quadratic equation and then remember to transform back to the original equation.
Factorable equations: Look for a factor common to all terms or factor by grouping.
Week 4 Day 2
APPLICATION PROBLEMS
Start
Read and analyze the problem
Make a diagram or sketch if possible
Determine the unknown quantity.
Did you set up the equation?
Set up an equation, assign variables to represent what you are asked to find.
Ano yes
A
Solve the equation
Check the solution
Is the unknown solved?
no
yes
End
RECALL Week 4 Day 3
APPLICATION PROBLEMS
1. If a person drops a water balloon off the rooftop of an 81 foot building, the height of the water balloon is given by the equation where t is in seconds. When will the water balloon hit the ground?
(Classroom example 1.3.12 page 122)
2. You have a rectangular box in which you can place a 7 foot long fishing rod perfectly on the diagonal. If the length of the box is 6 feet, how wide is that box?
(Classroom example 1.3.13 page 123)
3. A base ball diamond is a square. The distance from base to base is 90 feet. What is the distance from the home plate to the second base?
(#108 page 125)
Week 4 Day 3
81t16h 2
4. Lindsay and Kimmie, working together, can balance the financials for the Kappa Kappa Gama sorority in 6days. Lindsay by herself can complete the job in 5days less than Kimmie. How long will it take Lindsay to complete the job by herself? (# 113 page 125)
5.A rectangular piece of cardboard whose length is twice its width is used to construct an open box. Cutting a I foot by 1 foot square off of each corner and folding up the edges will yield an open box. If the desired volume is 12 cubic feet, what are the dimensions of the original piece of cardboard? (# 110 page 125)
6.Aspeed boat takes 1 hour longer to go 24 miles up a river than to return. If the boat cruises at 10mph in still water, what is the rate of the current? (#140 page 126)
Week 4 Day 3
7. Cost for health insurance with a private policy is given by where C is the cost per day and a is the insured’s age in years. Health
insurance for a six year old, a=6, is $4 a day (or $1,460 per year). At what age would someone be paying $9 a day (or $3,285 per year).
(#73 page 134)
8. The period (T) of a pendulum is related to the length (L) of the pendulum and acceleration due to gravity (g) by the formula
. If the gravity is and the period is 1 second find the approximate length of the pendulum. Round to the nearest inch. (#80 page 134)
Week 4 Day 3
a10C
gL
2T 2s/ft32
HOMEWORK
#s 8,31,44,53,56,66, 68,72,83,84,102, 104,106,114, 118,142 page124-127
#s 28, 50,72 page 133
Week 4 Day 3
