X factoring revised
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The document provides step-by-step instructions for factorizing a quadratic trinomial using the grouping method. The method involves: 1) drawing an X and writing the middle and first times last terms; 2) finding two factors whose sum is the middle term and product is the first times last; 3) writing the factors on either side of the X and dividing by the leading coefficient; 4) plugging the factors into the (x + )(x + ) format and moving the denominator in front of the x terms. When completed, the factorization is obtained.
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Final Exam Name___________________________________Si.docx
Final Exam Name___________________________________
Silva Math 96 Spring 2020
YOU MUST SHOW ALL WORK AND BOX YOUR ANSWERS FOR CREDIT. WORK ALONE.
Solve the absolute value inequality. Write your answer
in interval notation.
1) |2x - 12 |> 2
Solve the compound inequality. Graph the solution set.
Write your answer in interval notation.
2) -4x > -8 and x + 4 > 3
Solve the three-part inequality. Write your answer in
interval notation.
3) -1 < 3x + 2 < 14
Solve the absolute value equation.
4) 4x + 9 = 2x + 7
Solve the compound inequality.
5) 3( x + 4 ) ≥ 0 or 4 ( x + 4 ) ≤ 4
Solve the inequality. Graph the solution set and write
your answer in interval notation.
6) |5k + 8| > -6
Solve the inequality graphically. Write your answer in
interval notation .
7) x + 3 ≥ 1
x-8 -6 -4 -2 2
y
8
6
4
2
x-8 -6 -4 -2 2
y
8
6
4
2
1
Graph the system of inequalities.
8) 2x + 8y ≥ -4
y < - 3
2
x + 6
x-10 -8 -6 -4 -2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y
10
8
6
4
2
-2
-4
-6
-8
-10
Find the determinant of the given matrix.
9) 10 5
0 -4
Use Cramer's rule to solve the system of linear
equations.
10) 6x + 5y = -12
2x - 2y = -4
Write a system that models the situation. Then solve the
system using any method. Must show work for credit.
11)A vendor sells hot dogs, bags of potato chips,
and soft drinks. A customer buys 3 hot dogs,
4 bags of potato chips, and 5 soft drinks for
$14.00. The price of a hot dog is $0.25 more
than the price of a bag of potato chips. The
cost of a soft drink is $1.25 less than the price
of two hot dogs. Find the cost of each item.
Use row reduced echelon form to solve the system.
12) x + y + z = 3
x - y + 4z = 11
5x + y + z = -9
2
Find the domain of f. Write your answer in interval
notation.
13) f(x) = 13 - 9x
If possible, simplify the expression. If any variables
exist, assume that they are positive.
14) 2x + 6 32x + 6 8x
Match to the equivalent expression.
15) 100-1/2
A) 1
1000
B) 1
10
C) 1
100
D) 1
10
Write the expression in standard form.
16) (5 + 8i) - (-3 + i)
Simplify the expression. Assume that all variables are
positive.
17) 5 t
5
z10
Solve the equation.
18) 3x + 1 = 3 + x - 4
Write the expression in standard form.
19) 3 + 3i
5 + 3i
3
Write the equation in vertex form.
20) y = x2 + 5x + 2
The graph of ax2 + bx + c is given. Use this graph to solve
ax2 + bx + c = 0, if possible.
21)
x-5 5 10
y
50
40
30
20
10
-10
-20
-30
-40
-50
x-5 5 10
y
50
40
30
20
10
-10
-20
-30
-40
-50
Solve the equation. Write complex solutions in standard
form.
22) 4x2 + 5x + 5 = 0
Graph the quadratic function by its properties.
23) f(x) = 1
3
x2 - 2x + 3
x
y
x
y
Solve the equation. Find all real solutions.
24) 2(x - 1)2 + 11(x - 1) + 12 = 0
Solve the problem.
25) The length of a table is 12 inches more than its
width. If the area of the table is 2668 square
inches, what is its length?
4
Solve the equation..
Perfect square of Binomials
1. The square of a binomial (a + b) is a trinomial with terms a2, 2ab, and b2.
2. To square a binomial, square each term and multiply the unlike terms by 2.
3. Examples are provided of squaring binomials like (x + 6)2 = x2 + 12x + 36 and factoring trinomials into perfect square forms like (x - 2)2.
Solving quadratic equations
This document provides steps for solving various types of quadratic equations:
1) Factoring trinomials into two binomials and setting each binomial equal to zero to solve for x.
2) Solving basic square root problems by taking the square root of numbers or adding/subtracting and then taking the square root.
3) Setting regular trinomials equal to zero, factoring, and setting each binomial equal to zero to solve for x, which may require first adding/subtracting to group like terms.
Sample problems are provided with spaces to fill in the solutions for practice.
March 23, 2015
The document outlines a lesson on factoring polynomials, including reviewing factoring perfect square trinomials and introducing methods for factoring general trinomials of the form ax^2 + bx + c, such as grouping, the box method, and trial and error. It also notes an upcoming factoring test and provides examples and notes for students to complete as class work.
Quadratic equations class 10
The document discusses quadratic equations. It defines the standard form of a quadratic equation as ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The values of x that satisfy the equation are called the roots. A quadratic equation can have two distinct real roots, two equal real roots, or no real roots depending on the discriminant. The quadratic formula x = (-b ± √(b2 - 4ac))/2a is presented to directly obtain the roots from the standard form. The discriminant (D = b2 - 4ac) determines the nature of the roots. If D > 0, there are two distinct real roots. If D = 0,
Solving Quadratic Equations
This document discusses how to solve quadratic equations by factoring, using the quadratic formula, and determining the number and type of roots using the discriminant. Key steps include factoring the equation if possible to set each factor equal to 0 and solve, plugging the coefficients a, b, and c into the quadratic formula if factoring is not possible, and using the discriminant b^2 - 4ac to determine the number and type of roots. Examples are provided to demonstrate each method.
Mayank and Srishti presentation on gyandeep public school
This document discusses quadratic equations. It begins by thanking teachers for allowing students to do a project on quadratic equations. It then provides a brief history of quadratic equations and defines them as polynomial equations of degree 2 in the form of ax2 + bx + c = 0. It discusses roots, different forms quadratic equations can take, methods for solving them including factoring and the quadratic formula, and the concept of the discriminant. Examples are provided to illustrate solving by factoring and using the quadratic formula. In the end it provides sources used in the document.
Math final blog
This lesson plan introduces the quadratic function through its standard form of ax^2 + bx + c = 0 where a is not equal to zero. It discusses how to identify a quadratic function through its parabolic graph shape. The lesson then covers the three main methods for solving quadratics: factoring, completing the square, and using the quadratic formula. Factoring and completing the square are demonstrated through examples, while the quadratic formula is introduced through a memorable song.
factorization
This document discusses various methods for factoring polynomials, including:
1) Factoring trinomials by finding two numbers that multiply to the constant term and add to the middle coefficient.
2) The "box method" for factoring trinomials where the leading coefficient is not 1, which involves arranging terms in a 2x2 grid.
3) Factoring the difference of two squares using the formula (a + b)(a - b).
Solving Equations Transformable to Quadratic Equation Including Rational Alge...
Provides examples and solutions on how to solve equations that is transformable to quadratic equation and rational algebraic equations.
Factoring and Box Method
This document discusses various techniques for factoring polynomials, including:
1. Factoring using the greatest common factor (GCF).
2. Factoring polynomials with 4 or more terms by grouping.
3. Factoring trinomials using factors that add up to the coefficient of the middle term.
4. Using the "box method" to factor trinomials where the coefficient of the x^2 term is not 1.
5. Factoring the difference of two squares using the formula a^2 - b^2 = (a + b)(a - b).
Lesson plan final
This document provides instructions for teaching students about factoring quadratic trinomials. It explains that the product of two binomials with a common term, such as (a + b)(a + c), can be expressed by the formula a2 + (b + c)a + bc. This formula results in a quadratic trinomial since it has three terms. The factors of the trinomial are simply the reverse of this formula. Students are guided through examples of factoring various quadratic trinomials and then do a group activity to practice factoring more examples. They then present their work, followed by a quiz and assignment to further their understanding.
Dev
The document describes three math problems:
1) A landscaper needs to lay the largest possible basketball court within a budget. The optimal dimensions are found to be 32.66 feet wide by 57.15 feet long.
2) Simplifying a fraction involving polynomials by factoring and canceling terms.
3) Solving a quadratic equation by completing the square. The solutions are found to be -4.7 and -0.3.
Chithra
This document provides an overview of different methods for solving quadratic equations: factorisation, completing the square, and the quadratic formula. It includes examples of solving quadratic equations using each method. Factorisation involves finding two binomial factors whose product is the quadratic expression. Completing the square transforms the equation into a perfect square plus an extra term, allowing it to be factorised. The quadratic formula provides the general solution for any quadratic equation in the form ax^2 + bx + c = 0.
Question 1 1. Evaluate using integration by parts. l.docx
Question 1
1.
Evaluate using integration by parts.
ln x - + C
x2 ln x - + C
x2 ln x - x2 + C
x ln x - x + C
2 points
Question 2
1.
Evaluate using integration by parts.
x2e2x - xe2x + e2x + C
x2e2x - xe2x + e2x + C
x2e2x - xe2x + C
x2e2x - xe2x + e2x + C
2 points
Question 3
1.
Evaluate. Assume u > 0 when ln u appears.
dy
2e4y + C
e4y + C
4e4y + C
e4y + C
2 points
Question 4
1.
Find the particular solution determined by the given condition.
f'(x) = 6x2 - 4x + 21; f(1) = 17
f(x) = 2x3 - 4x2 + 21x - 2
f(x) = 6x3 - 4x2 + 21x - 6
f(x) = 2x3 - 2x2 + 21x + 4
f(x) = 2x3 - 2x2 + 21x - 4
2 points
Question 5
1.
Find the particular solution determined by the given condition.
y' = ; y = 21 when x = 1
y = 5 ln x + 21
y = ln x + 19
y = 5 ln x + 2.5
y = ln x + 21
2 points
Question 6
1.
Determine if the function is a solution to the given differential equation.
y = x ln x - 4x + 5; y'' - = 0.
Yes
No
2 points
Question 7
1.
Evaluate using integration by parts.
(x2 - x) ln (16x) - + 2x + C
(x2 - x) ln (16x) - + x + C
ln (16x) - + x + C
(x2 - x) ln (16x) - x2 + x + C
2 points
Question 8
1.
Find the general solution for the differential equation.
= 4P
P = 4eCt
P = Ce4t
P = Ce-4t
P = Cet
2 points
Question 9
1.
Evaluate. Assume u > 0 when ln u appears.
dt
e-7t2 + C
- e-7t2 + C
e-7t2 + C
- e-7t2 + C
2 points
Question 10
1.
Find the general solution for the differential equation.
y ' = 72x2 - 20x
72x3 - 20x2 + C
24x3 - 20x2 + C
72x3 - 10x2 + C
24x3 - 10x2 + C
2 points
Question 11
1.
Find the general solution for the differential equation.
y ' = x - 16
2x2 - 16 + C
- 16x + C
- x + C
x3 - 16x + C
2 points
Question 12
1.
Evaluate using integration by parts.
e4xdx
(x - 8) e4x - e4x + C
(x - 8) e4x + e4x + C
4(x - 8) e4x - 16 e4x + C
(x - 8) e4x - e4x + C
2 points
Question 13
1.
Evaluate using integration by parts.
dx
5x(2x + 3)1/2 + (2x + 3)3/2 + C
5x(2x + 3)1/2 - (2x + 3)3/2 + C
x(2x + 3)1/2 - (2x + 3)3/2 + C
5x(2x + 3)1/2 - (2x + 3)3/2 + C
2 points
Question 14
1.
Write the first four elements of the sequence.
n
0, 2, ,
2, , ,
1, , ,
0, 1, ,
2 points
Question 15
1.
Evaluate. Assume u > 0 when ln u appears.
dx
+ C
+ C
+ C
(ln 6x)2 + C
2 points
Question 16
1.
Evaluate. Assume u > 0 when ln u appears.
dp
e5p2+ C
- e5p2 + C
7e5p2 + C
-7e5p2 + C
2 points
Question 17
1.
Evaluate. Assume u > 0 when ln u appears.
dy
ln + C
18 ln + C
19 ln + C
ln + C
2 points
Question 18
1.
Evaluate using integration by parts.
dx
x(ln 2x)2 - 2 ln (2x) + C
ln (2x)2 - 2x ln (2x) - 2x + C
x(ln 2x)2 + 2x ln (2x) + 2x + C
x(ln 2x)2 - 2x ln (2x) + 2x + C
2 points
Question 19
1.
Evaluate using integration by parts.
ln x dx
x2ln x - x2 + C
ln x - x2 + C
x2ln x - x2 + 5x + C
x2ln x - x2 - 5x + C
2 ...
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X factoring revised
- 1. WARM UPS Multiply by distribution or FOIL method 1) (x + 2)(x + 4) 2)(x – 3)(2x + 4) 3) x2 – x – 6 4) x2 + 7x + 12 Factor using grouping method
- 2. California Standards 14.0 Students solve a quadratic equation by factoring or completing the square. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
- 3. ax2 + bx + cStep 1: Draw an X
- 4. ax2 + bx + cStep 2: Fill in the “X” with the middle term, b… And the 1st term times the last terms, or a • c b a• c
- 5. ax2 + bx + cStep 3: Find two “things” or numbers that product= a• c And their Sum = b Put these things, one on each side of X. b a• c = # = #
- 6. ax2 + bx + cStep 4: Divide the numbers that you find that have a product =a• c And sum = b by a. REDUCE THE FRACTIONS! b a• c _#_ a _#_ a
- 7. ax2 + bx + cStep 5: Plug the terms On the sides of the big X into the (X + _)(X + _) Format b a • c _#_ a (X + _#_ ) (X + _#_ ) a a _#_ a
- 8. ax2 + bx + cStep 6: Move the denominator in front of the X for both terms. This is your factorization of the trinomial (X + _#_ ) (X + _#_ ) a a (aX + _#_ ) (aX + _#_ ) NOTE that the number in front of the x may not equal a since we reduced the fraction in step 5.
- 9. We don’t get it…can you show us?
- 10. 3x2 - 10x + 8Step 1: Draw an X
- 11. Step 2: Fill in the “X” with the middle term, -10… And the 1st term times the last terms, or 3 • -8 -10 3 • -8 = -24 3x2 - 10x - 8 -24
- 12. Step 4: Find two “things” such that their sum is -10 and their product is -24 Thing 1 + Thing 2 = -10 Thing 1 • Thing 2 = -24 3x2 - 10x - 8 -10 -24 Thing 1 (#) Thing 2 (#)
- 13. Step 4 (continued): Find two “things” such that their sum is -10 and their product is 24 Try all the factors of -24 until you get the correct one -1 x 24 = 25 no 2 x -12 = 24 yes 2 + (-12) = -10 yes -12 3x2 - 10x - 8 -10 -24 2
- 14. -12= -4 3x2 – 10x - 24 -10 -24 2 Step 5: Now divide both sides by a, or 3. Reduce!!!! Now plug into (x + _)(x + _) 3 3 (X + _-4_ ) (X + _2_ ) 1 3 1
- 15. ax2 + bx + cStep 6: Move the denominator in front of the X for both terms. This is your factorization of the trinomial (X - _4_ ) (X + _2_ ) 1 3 (1X - 4 ) (2X + 3 ) NOTE that the number in front of the x may not equal a since we reduced the fraction in step 5.
- 16. The factors of are 3x2 - 10x - 24 (x – 4) (3x + 2)
- 17. Ok, using a thinking map to summarize… Draw X Write middle term Write first • last middle middle First • Last Find factors of a x c, sum of b Write things (factors) on sides Divide numbers by a, then reduce Find two “things” such that their sum is the “middle” and their product is the “first • last” ax2 + bx + c middle First • Last ? ? middle First • Last # # Plug into (x+ # /a) (x+ # /a) (#/a is reduced) (ax + #) (ax + #) middle First • Last Move reduced a in front of x and your done!