Answer:
The first term is, x2 its coefficient is 1 .
The middle term is, -17x its coefficient is -17 .
The last term, "the constant", is +60
Step-1 : Multiply the coefficient of the first term by the constant 1 • 60 = 60
Step-2 : Find two factors of 60 whose sum equals the coefficient of the middle term, which is -17 .
-60 + -1 = -61
-30 + -2 = -32
-20 + -3 = -23
-15 + -4 = -19
-12 + -5 = -17 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -12 and -5
x2 - 12x - 5x - 60
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-12)
Add up the last 2 terms, pulling out common factors :
5 • (x-12)
Step-5 : Add up the four terms of step 4 :
(x-5) • (x-12)
Which is the desired factorization
Equation at the end of step
1
:
(x - 5) • (x - 12) = 0
STEP
2
:
Theory - Roots of a product
2.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation:
2.2 Solve : x-5 = 0
Add 5 to both sides of the equation :
x = 5
Solving a Single Variable Equation:
2.3 Solve : x-12 = 0
Add 12 to both sides of the equation :
x = 12
Supplement : Solving Quadratic Equation Directly
Solving x2-17x+60 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula
Parabola, Finding the Vertex:
3.1 Find the Vertex of y = x2-17x+60
For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 8.5000
Plugging into the parabola formula 8.5000 for x we can calculate the y -coordinate :
y = 1.0 * 8.50 * 8.50 - 17.0 * 8.50 + 60.0
or y = -12.250
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = x2-17x+60
Axis of Symmetry (dashed) {x}={ 8.50}
Vertex at {x,y} = { 8.50,-12.25}
x -Intercepts (Roots) :
Root 1 at {x,y} = { 5.00, 0.00}
Root 2 at {x,y} = {12.00, 0.00}
Solve Quadratic Equation by Completing The Square
3.2 Solving x2-17x+60 = 0 by Completing The Square .
Subtract 60 from both side of the equation :
x2-17x = -60
Now the clever bit: Take the coefficient of x , which is 17 , divide by two, giving 17/2 , and finally square it giving 289/4
Add 289/4 to both sides of the equation :
On the right hand side we have :
-60 + 289/4 or, (-60/1)+(289/4)
The common denominator of the two fractions is 4 Adding (-240/4)+(289/4) gives 49/4
So adding to both sides we finally get :
x2-17x+(289/4) = 49/4
Adding 289/4 has completed the left hand side into a perfect square :
x2-17x+(289/4) =
(x-(17/2)) • (x-(17/2)) =
(x-(17/2))2
Things which are equal to the same thing are also equal to one another. Since
x2-17x+(289/4) = 49/4 and
x2-17x+(289/4) = (x-(17/2))2
then, according to the law of transitivity,
(x-(17/2))2 = 49/4
We'll refer to this Equation as Eq. #3.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(17/2))2 is
(x-(17/2))2/2 =
(x-(17/2))1 =
x-(17/2)
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x-(17/2) = √ 49/4
Add 17/2 to both sides to obtain:
x = 17/2 + √ 49/4
Since a square root has two values, one positive and the other negative
x2 - 17x + 60 = 0
has two solutions:
x = 17/2 + √ 49/4
or
x = 17/2 - √ 49/4
Note that √ 49/4 can be written as
√ 49 / √ 4 which is 7 / 2
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving x2-17x+60 = 0 by the Quadratic Formula .
According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
x = ————————
2A
In our case, A = 1
B = -17
C = 60
Accordingly, B2 - 4AC =
289 - 240 =
49
Applying the quadratic formula :
17 ± √ 49
x = —————
2
Can √ 49 be simplified ?
Yes! The prime factorization of 49 is
7•7
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
√ 49 = √ 7•7 =
± 7 • √ 1 =
± 7
So now we are looking at:
x = ( 17 ± 7) / 2
Two real solutions:
x =(17+√49)/2=(17+7)/2= 12.000
or:
x =(17-√49)/2=(17-7)/2= 5.000
Two solutions were found :
x = 12
x = 5
Step-by-step explanation:
please mark my answer in brainlist
need help with these two math problems
Answer:
y = (x - 1)(x - 3)
y = (x + 2)^2
Step-by-step explanation:
3)
Has roots at x = {1, 3}
y = (x - 1)(x - 3)
4)
Has double root at x = -2
y = (x + 2)^2
Find the equation of the line passing through the points 7,2 and 9,1
9514 1404 393
Answer:
y = -1/2x +11/2
Step-by-step explanation:
The slope of the line is ...
m = (y2 -y1)/(x2 -x1)
m = (1 -2)/(9 -7) = -1/2
The y-intercept is ...
b = y -mx
b = 2 -(-1/2)(7) = 11/2
Then the slope-intercept equation is ...
y = -1/2x +11/2
_____
Alternative solution
A general form equation for the line can be ...
(y1 -y2)(x -x1) -(x1 -x2)(y -y1) = 0
(2 -1)(x -7) -(7 -9)(y -2) = 0
x-7 +2y -4 = 0
x +2y -11 = 0 . . . . . general form equation
x +2y = 11 . . . . . . . standard form equation
Note that we want the x-coefficient to be positive, so we chose the order of the points to make that be the case.
Suzanne correctly answered 32 of the 40 questions on her math test
What decimal is equivalent to the fraction of questions Suzanne answered correctly?
Enter your answer in the box.
Answer:
Step-by-step explanation: 50 million
43 is 18% of what number?
Answer:
238.888888889
Step-by-step explanation:
Divide 43 by 18, then multiply by 100
Please help asap thank you!!
Answer:
No
yes
No
Step-by-step explanation:
yes it is a scalene triangle
the area of the triangle is 28.5
PLEASE NO LINKS DONT WASTE YOUR TIME IM NOT GONNA CLICK. Use the graph of the polynomial function to find the factored form of the
related polynomial. Assume it has no constant factor.
Answer:
B.
Step-by-step explanation:
factored form actually contains the x intercepts right in it. Basically whatever sets the whole equation equal to zero.
On the graph you can see that it passes through both positive and negative 4. so (x+4)(x-4) is the answer because plugging in 4 for x will make it zero. Same for plugging in negative 4
Hope this helps!
When you are multiplying like bases with exponents, you will also multiply the exponents together and leave the same base in your solution.
TRUE or FALSE?
come on now
[tex]\underset{\textit{like-bases with exponents}}{5^{11}\cdot 5^3\cdot 5^{17}}\implies \stackrel{\textit{add the exponents}}{\underset{\textit{keep the base}}{5^{11+3+17}}}\implies 5^{31}[/tex]
Maria wrote her 606 word
essay in 20 1/5 minutes. At
what rate is she typing?
Answer:
29.5
Step-by-step explanation:
Multiply 606 and 20.5 and get 29.5.
3/4 x 2 5/6 ÷ 2 + 34
Answer: 35 1/16
Exact form: 561/16
Explanation:
If the legs of each triangular base are 5 cm and 3 cm and the height of the prism is 4 cm, what
are the coordinates of the vertices of the prism?
The coordinates of the six vertices of the prism are listed below:
[tex](x_{1},y_{1}, z_{1}) = (0, 0, 0)\,[ft][/tex] [tex](x_{2}, y_{2}, z_{2}) = (0, 0, 5)\,[ft][/tex] [tex](x_{3}, y_{3}, z_{3}) = (3, 0, 0)\,[ft][/tex] [tex](x_{4}, y_{4}, z_{4}) = (0, 4, 5)\,[ft][/tex] [tex](x_{5}, y_{5}, z_{5}) = (0, 4, 0)\,[ft][/tex][tex](x_{6}, y_{6}, z_{6}) = (3, 4, 0)\,[ft][/tex]The number of vertices of a prism with a polygonal base ([tex]n[/tex]) is described by the following expression:
[tex]n = 2\cdot s[/tex] (1)
Where [tex]s[/tex] is the number of sides of the base.
If we know that [tex]s = 3[/tex], then the number of vertices of the prism are:
[tex]n = 2\cdot 3[/tex]
[tex]n = 6[/tex]
A prism with triangular base has six vertices, let suppose that the triangle of the base is right-angled. and coincides with a right-hand rectangular coordinate system. Then, we can conclude that the coordinates of the six vertices are the following:
[tex](x_{1},y_{1}, z_{1}) = (0, 0, 0)\,[ft][/tex] [tex](x_{2}, y_{2}, z_{2}) = (0, 0, 5)\,[ft][/tex] [tex](x_{3}, y_{3}, z_{3}) = (3, 0, 0)\,[ft][/tex] [tex](x_{4}, y_{4}, z_{4}) = (0, 4, 5)\,[ft][/tex] [tex](x_{5}, y_{5}, z_{5}) = (0, 4, 0)\,[ft][/tex][tex](x_{6}, y_{6}, z_{6}) = (3, 4, 0)\,[ft][/tex]We kindly invite to check this question on prisms: https://brainly.com/question/8828506
What is the solution to x + 12 = 25?
A. x=13
B. x=12
C. x=37
D. x=24
Answer:
A . x=13 is the solution of this question.
Answer:
x=3
Step-by-step explanation:
subtract 12 from both sides of the equation
x+12=25
x+12-12=25-12
then subtract the numbers
x=13
if 1/3 of a gallon of paint covers 1/8 how much paint is needed to cover the entire wall
Answer:
4 gallons
Step-by-step explanation:
Your friend tells you that at 1:00 pm the temperature outside was 67 degrees Fahrenheit and at 5:00 pm, that same day, the temperature was 65 degrees Fahrenheit. He also states that the average rate of change of the temperature between 1:00 pm and 5:00 pm increased by 2 degrees Fahrenheit. Determine if your friend is correct ?
Answer:
Incorrect
Step-by-step explanation:
Average rate of change is
(65 - 67) / (5:00 - 1:00) = -½°/hr
Two decimal for 2597.58333
Answer:
2597. 58000 and this is the ans
SUBTRACT the given polynomials : (x + 2) - (8x + 4)
A}7x + 2
B}7x - 2
C}-7x - 2
D}-7x + 2
WELP ;-;
The equivalent expression for the subtraction (x + 2) - (8x + 4) is -7x - 2
Option C is correct
The given polynomial expression is:
(x + 2) - (8x + 4)
Expand the expression by removing the parenthesis
x + 2 - 8x - 4
Collect like terms
x - 8x + 2 - 4
-7x - 2
Therefore, the equivalent expression for the subtraction (x + 2) - (8x + 4) is -7x - 2
Learn more on polynomials here:
https://brainly.com/question/16058963
Write the product as a power.
18 x 18 x 18 x 18 =
Answer:
[tex]18^{4}[/tex]
Step-by-step explanation:
You multiply it by itself 4 times, therefore being 18 to the power of 4
Answer:
[tex]18^{4}[/tex]
Step-by-step explanation:
A power is simply how many times a number is multiplied by itself. Since the question is 18 multiplied by itself four times (count the number of 18s), the power is 4.
What is a common denominator for 1 3/4 and 1 4/5
Answer:
20
Step-by-step explanation:
A common denominator is a number that is a multiple of both denominators.
To find it just write the multiplication table of the numbers and find a number that appears in both of them.
In this case, we have 4 and 5:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40
5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Since it only asks for a common denominator, it can be 20, 40, 60, 80... Although it's usually smarter to pick the lower number.
[tex]1\dfrac 34 =1+ \dfrac 34 =\dfrac 74 = \dfrac{35}{20}\\\\\\1\dfrac 45 = 1+ \dfrac 45 = \dfrac 95 =\dfrac{36}{20}\\\\\text{The answer is 20}[/tex]
Kenrick mows lawns on Sundays. Last
week it took him 5 hours to finish.
This week it took only 5 hours. How
many times longer did it take last
week than this week?
Answer:
0 hours.
Step-by-step explanation:
He finished at the same time than Kenrick finished last week. So, he/she did not delay in mowing the lawn. Hoped this helped.
the sum of any 2 intergers is either postive or negative. true or false
Answer:
True.
Hope this helps.
Calculations with Percent - Item 35698
Question 7
Asha bought 60 fruit baskets.
25% of the fruit baskets have 12 pieces of fruit in each basket.
The remaining 75% of the baskets have 15 pieces of fruits in each basket.
Complete the model.
25%
75%
Number of baskets with 12 piece of
fruit:
Number of baskets with 15 pieces of fruit:
60 fruit baskets
The number of baskets with 12 piece of fruit is 15.
The number of baskets with 15 pieces of fruit is 45.
Percentage is the fraction out of a number expressed as a number out of hundred. The sign used to represent percentages is %.
In order to determine the number of baskets that have 12 pieces of fruits and 15 pieces of fruits, multiply the given percentages by the total number of fruit baskets.
Number of baskets with 12 piece of fruit: 25% x 60 = 0.25 x 60 = 15
Number of baskets with 15 pieces of fruit: 75% x 60 = 0.75 x 60 = 45
To learn more about percentages, please check: https://brainly.com/question/92258
How much is f(x) = 3(4)x
Answer:
f ( x ) = 12 x
Step-by-step explanation: 3x4=12 so that makes it f(x)=12x
Multiply.
3 1/2 times 2 2/3
Answer:
Step-by-step explanation:
help plz LCM/GCF plz help with 2, 4, and 5 show your work for brainliest
THIS IS WORTH 100 POINTS!!!
Answer:
2. 60
4. 6
5. 3
Step-by-step explanation:
These questions are all asking for the GCF, which is the greatest common factor, or the largest number that can be divided into all of the numbers.
ex. for 2, the question is basically asking for the GCF between 12 and 15
we see that 12 = 3x4, 15 = 3x5, so from here we see that 3 is the GCF.
Solve by elimination Solve by elimination or substitution: 4x + 2y = 6 and x - 3y = 5
Step-by-step explanation:
since we need to multiply only one equation with something, elimination might be a good method here. we bring both equations to 4x terms and then subtract the second from the first equation :
4x + 2y = 6
- 4x - 12y = 20
-----------------------
0 14y = -14
y = -1
=> e.g. in the second original equation
x - 3×-1 = 5
x + 3 = 5
x = 2
Choose the end behavior of the graph of each polynomial function
Answer:
D, A, B
Step-by-step explanation:
The sign of the leading coefficient always tells you the end behavior on the right: positive = rises; negative = falls.
The degree of the polynomial tells you how the left- and right-end behaviors compare: even = they are the same; odd = they are opposites.
__
A) negative leading coefficient, even degree:
falls to the left, falls to the right (D)__
B) positive leading coefficient, odd degree:
falls to the left, rises to the right (A)__
C) negative leading coefficient, odd degree
rises to the left, falls to the right (B)Please help with give branniest!!
(11/44)^ 3 = 11^3/14^3 = 1331/2744 = 0.48505
Round the answer as needed.
Which is the best estimate of the distance between the point (–25, 10) and the origin? 8 units 15 units 23 units 27 units
Answer:
27 units
Step-by-step explanation:
I don't think that diagonals count so I think it is 27 units.
Tell me if I'm wrong :)
I NEED HELP PLSSSSSS
Answer:
130 is the answer I Think
Which
graph shows the solution to the
system of linear inequalities?
y > 2/3x + 3
y < - 1/3x + 2
-
Answer:there is no graphs
Step-by-step explanation:
poop
Simplify the expression ((1/2)^3)^3
Helppp