The document discusses solving quadratic equations by factoring. It begins with an example problem and explanation of the zero factor property. It then outlines the steps to solve quadratic equations by factoring: 1) write the equation in standard form, 2) factor completely, 3) set each factor equal to 0, 4) solve each equation, and 5) check solutions. Several example problems are worked through demonstrating this process. The document concludes with examples of using factoring to solve real-world problems involving quadratic equations.
* Find zeros of polynomial functions
* Use the Fundamental Theorem of Algebra to find a function that satisfies given conditions
* Find all zeros of a polynomial function
* Find zeros of polynomial functions
* Use the Fundamental Theorem of Algebra to find a function that satisfies given conditions
* Find all zeros of a polynomial function
Question 11. Determine which of the following points lies on .docxmakdul
Question 1
1. Determine which of the following points lies on the graph of the equation:
(0,7)
(0,6)
(0,5)
(6,5)
(1,5)
Question 2
Complete the table. Use the resulting solution points to sketch the graph of the equation.
Question3
Graphically estimate the x- and y- intercepts of the graph:
y = x3 - 9x
x-intercept: (±3,0),(0,0)
y-intercept: (0,0)
x-intercept: (3,0),(0,0)
y-intercept: (0,0)
x-intercept: (-3,0),(0,0)
y-intercept: (0,0)
x-intercept: (0,±3),(0,0)
y-intercept: (0,0)
x-intercept (0,3),(0,0)
y-intercept (0,0)
Question 4
Find the x- and y-intercepts of the graph of the equation
y=49-7x
x-intercept: (7,0)
y-intercept: (0,-7)
x-intercept: (49,0)
y-intercept: (0,7)
x-intercept: (-7,0)
y-intercept: (0,-49)
x-intercept: (49,0)
y-intercept: (0,49)
x-intercept: (7,0)
y-intercept: (0,49)
Question 5
Determine whether the value of x=7 is a solution of the equation:
2.
no
yes
Question 6
Solve the equation 8-5x=6
Question 7
Solve the equation and check your solution.
-2-4x=30
9
-11
-8
7
-10
Question 8
Solve the equation and check your solution.
5y + 1 = 6y - 5 + 8y
2/3
3/2
6/5
5/6
-2/3
Question 9
Solve the equation and check your solution.
67x - 24 = 3x + 8(8x-3)
3
67
-3
-67
All real numbers
Question 10
Solve the equation and check your solution.
10
6
7
9
8
Question 11
Solve the equation and check your solution. (If not possible, explain why.)
2
5
6
4
10
Question 12
3.
4. Solve the equation and check your solution. (If not possible, explain why)
5.
-18
7
11
No solution. The variable is divided out.
20
Question 13
Write the quadratic equation in general form.
4x2 = 8 - 9x
4x2 + 9x + 8 = 0
4x2 + 9x = -8
4x2 - 9x - 8 = 0
-4x2 + 9x - 8 = 0
4x2 + 9x - 8 = 0
Question 14
Solve the quadratic equation by factoring.
x2 - 6x + 5 = 0
-1, 5
-1,-5
1,-5
1,5
6,5
Question 15
Solve the quadratic equation by factoring.
x2 + 8x + 16 = 0
4
-1/4
-4
±4
1/4
Question 16
Solve the equation by extracting square roots.
(x+6)2 = 5
6 + √5
-6 ± √5
-6 -√5
6 ± √5
-6 + √5
Question 17
Use the Quadratic Formula to solve
x2 + 20x + 98 = 0
x = -8, x = -12
x = -√2 - 10, x = √2 - 10
x = -√3 - 10, x = √3 - 10
x = 10, x = -10
x = -√2 - 9, x = √2 - 9
Question 18
Write the complex number in standard form.
√ -9
3i
-3i
9i
4i
-9i
Question 19
Find real numbers a and b such that the equation is true.
a + bi = 14 + 2i
a=16, b=4
a=18, b=6
a=14, b=2
a=15, b=14
a=17, b=5
Question 20
Find all solutions to the following equation.
x = -17/4
x=9
no solution
x=-17
x=-8
Question 1
1. Determine which ...
College algebra in context 5th edition harshbarger solutions manualAnnuzzi19
College Algebra in Context 5th Edition Harshbarger Solutions Manual
Full download:https://goo.gl/eJxUd2
People also search:
college algebra in context with applications for the managerial life and social sciences
pearson
chegg
4. 11.6 – Solving Quadratic Equations by Factoring
A quadratic equation is written in the Standard
Form, ax2 bx c 0
where a, b, and c are real numbers and . 0 a
Examples:
2 x 7x 12 0
xx 7 0
2 3x 4x 15
(standard form)
5. 11.6 – Solving Quadratic Equations by Factoring
Zero Factor Property:
If a and b are real numbers and if , 0 ab
then or . 0 a 0 b
Examples:
xx 7 0
x 0 x 7 0
x 0 x 7
6. 11.6 – Solving Quadratic Equations by Factoring
Zero Factor Property:
If a and b are real numbers and if , 0 ab
then or . 0 a 0 b
Examples:
x103x6 0
x 10 0 3x6 0
x 1010 010 3x 66 06
x 10
x
3 6
3 3
3x 6 x 2
7. 11.6 – Solving Quadratic Equations by Factoring
Solving Quadratic Equations:
1) Write the equation in standard form.
2) Factor the equation completely.
3) Set each factor equal to 0.
4) Solve each equation.
5) Check the solutions (in original equation).
9. 11.6 – Solving Quadratic Equations by Factoring
If the Zero Factor
Property is not used,
then the solutions will
be incorrect
2 x 3x 18
xx3 18
x 18
2
18 3 18 18
32454 18
270 18
2
21 3 21 18
4416318
378 18
x318
x33183
x 21
10. 11.6 – Solving Quadratic Equations by Factoring
xx 4 5
x2 4x 5
2 x 4x 5 0
x 1x 5 0
x1 0 x 5 0
x 1 x 5
11. 11.6 – Solving Quadratic Equations by Factoring
x 33x 2 0
3x2 7x 6 30x 3 2 0 x
3 x
2
3
x
x3x 7 6
2 3 7 6 0 x x 32x
Factors of 3:
1, 3
Factors of 6:
1, 6 2, 3
12. 11.6 – Solving Quadratic Equations by Factoring
9x2 24x 16
2 9x 24x 16 0
9 and 16 are perfect squares
3x 43x 4 0
3x4 0
3x 4
4
3
x
13. 11.6 – Solving Quadratic Equations by Factoring
2x3 18x 0
2x
2x
2 9 x 0
3 x 3 x 0
x 3 0 x3 0
2x 0
x 0 x 3
x 3
14. 11.6 – Solving Quadratic Equations by Factoring
x 33x2 20x 7 0
3: Factors of 1, 37 : Factors of 1, 7
x3
30 x
7 x 0 3 1 x
70x 3 1 0 x
x 3 x 7
3x 1
1
3
x
15. Continuous assessment
• Q1: Solve these equations by factorisation.
a) (x-2)(2x+1) = 0
b) 3x2 – 27x = 0
c) 2x2 – 7x + 6 = 0
Q2: a) 2(d2 – 3d +3) = d + 1
b) 3(e + 1)2= 1 – e
c) (g + 3)(2 – g) = g2
16. Answers
• Q1: a) x = 2, x = - ½ b) x = 0, x= 9 c) x = -4, x=3
• Q2: a) (a+3)(a-3) b) e = -2, e = -1/3
c) d = -2, d = 1 ½
19. 11.7 – Quadratic Equations and Problem Solving
A cliff diver is 64 feet above the surface of the
water. The formula for calculating the height (h)
of the diver after t seconds is: h 16t2 64.
How long does it take for the diver to hit the
surface of the water?
2 16t 64
16 2 t 4
16 2 t 2 t
0
0
0
t 2 0 t 2 0
t 2 t 2 seconds
20. 11.7 – Quadratic Equations and Problem Solving
The square of a number minus twice the number
is 63. Find the number.
x is the number.
2 x
2x 63
2 x 2x 63 0
63: Factors of 1, 63 3, 21 7, 9
x 7
9 x 0
x 7 0 x 9 0
x 7
x 9
21. 11.7 – Quadratic Equations and Problem Solving
The length of a rectangular garden is 5 feet more than
its width. The area of the garden is 176 square feet.
What are the length and the width of the garden?
l wA
The width is w.
The length is w+5. w5w176
11 w
w11 0
w11
2 w 5w 176
2 w 5w176 0
w16 0
w 16
w11 l 115
l 16
feet
feet
Factors of 176:
1,176 2, 88 4, 44
8, 22 11,16
16 w 0
22. 11.7 – Quadratic Equations and Problem Solving
Find two consecutive odd numbers whose product is
23 more than their sum?
Consecutive odd numbers: x
x
2. x
5x 5x 2 2 2 25 x x x
2 x 25
2 x 25 0
x5
50 x 50 x
52 3 5 2 7
5, 3 5, 7
2 x 2 x x 23
2 x 2x 2x 2x 25 2x
2 x 25 25 25
x5 0
23. 11.7 – Quadratic Equations and Problem Solving
The length of one leg of a right triangle is 7 meters less than
the length of the other leg. The length of the hypotenuse is
13 meters. What are the lengths of the legs?
ax
a 12
Pythagorean Th.
2 2 2 x x 7 13
x 5
meters
5
7bx 13 c
2 2 x x 14x 49 169
2 2x 14x 120 0
2 2 x 7x 60 0
2
50x 12 0 x
x 12
b 127 meters
2 2 2 a b c
Factors of 60:1, 60 2, 30
3, 20 4,15 5,12
5 x 12 x 0
6,10