Answer:
-2,7
Step-by-step explanation:
[tex]\huge\bold\red{Answer:-}[/tex]
Step by step solution :
STEP 1 : Trying to factor by splitting the middle term
1.1 Factoring x2+5x-14
The first term is, x2 its coefficient is 1 .
The middle term is, +5x its coefficient is 5 .
The last term, "the constant", is -14
Step-1 : Multiply the coefficient of the first term by the constant 1 • -14 = -14
Step-2 : Find two factors of -14 whose sum equals the coefficient of the middle term, which is 5 .
-14 + 1 = -13
-7 + 2 = -5
-2 + 7 = 5 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -2 and 7
x2 - 2x + 7x - 14
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-2)
Add up the last 2 terms, pulling out common factors :
7 • (x-2)
Step-5 : Add up the four terms of step 4 :
(x+7) • (x-2)
Which is the desired factorization.
Equation at the end of step 1 :
(x + 7) • (x - 2) = 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+7 = 0
Subtract 7 from both sides of the equation :
x = -7
Solving a Single Variable Equation:
2.3 Solve : x-2 = 0
Add 2 to both sides of the equation :
x = 2
Supplement : Solving Quadratic Equation Directly
Solving x2+5x-14 = 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+5x-14
For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is -2.5000
Plugging into the parabola formula -2.5000 for x we can calculate the y -coordinate :
y = 1.0 * -2.50 * -2.50 + 5.0 * -2.50 - 14.0
or y = -20.250
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = x2+5x-14
Axis of Symmetry (dashed) {x}={-2.50}
Vertex at {x,y} = {-2.50,-20.25}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-7.00, 0.00}
Root 2 at {x,y} = { 2.00, 0.00}
Solve Quadratic Equation by Completing The Square
3.2 Solving x2+5x-14 = 0 by Completing The Square .
Add 14 to both side of the equation :
x2+5x = 14
Now the clever bit: Take the coefficient of x , which is 5 , divide by two, giving 5/2 , and finally square it giving 25/4
Add 25/4 to both sides of the equation :
On the right hand side we have :
14 + 25/4 or, (14/1)+(25/4)
The common denominator of the two fractions is 4 Adding (56/4)+(25/4) gives 81/4
So adding to both sides we finally get :
x2+5x+(25/4) = 81/4
Adding 25/4 has completed the left hand side into a perfect square :
= x2+5x+(25/4)
= (x+(5/2)) • (x+(5/2))
= (x+(5/2))2
Things which are equal to the same thing are also equal to one another. Since
x2+5x+(25/4) = 81/4 and
x2+5x+(25/4) = (x+(5/2))2
then, according to the law of transitivity,
(x+(5/2))2 = 81/4
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+(5/2))2 is
= (x+(5/2))2/2
= (x+(5/2))1
=x+(5/2)
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x+(5/2) = √ 81/4
Subtract 5/2 from both sides to obtain:
x = -5/2 + √ 81/4
Since a square root has two values, one positive and the other negative
x2 + 5x - 14 = 0
has two solutions:
x = -5/2 + √ 81/4
or
x = -5/2 - √ 81/4
Note that √ 81/4 can be written as
√ 81 / √ 4 which is 9 / 2
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving x2+5x-14 = 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 = 5
C = -14
Accordingly, = B2 - 4AC
=25 - (-56)
= 81
Applying the quadratic formula :
-5 ± √ 81
x = —————
2
Can √ 81 be simplified ?
Yes! The prime factorization of 81 is
3•3•3•3
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).
√ 81 = √ 3•3•3•3 =3•3•√ 1
= ± 9 • √ 1
= ± 9
So now we are looking at:
x = ( -5 ± 9) / 2
Two real solutions:
x =(-5+√81)/2=(-5+9)/2= 2.000
or:
x =(-5-√81)/2=(-5-9)/2= -7.000
Two solutions were found :
x = 2 x = -7Find the product. −3y2(−5y3 + y − 6)
Generalize the patterns you noticed in part D to create a rule or identity to describe those patterns. For example, if you notice that every time you multiply a negative number by another negative number the result is positive, we can generalize this by saying (-a)(-b) = c, where a, b, and c are all positive real numbers.
I know it is probably to late for your answer but if you are on Solving and Reasoning with Complex Numbers: Tutorial in Edmentum, then the answer for part D on the first lesson activity is this word for word.
This equation generalizes the patterns seen in part D, where a and b represent real numbers:
(a + bi)(a − bi) = a2 + b2.
Hope this helps good luck
John sells frozen fruit bars at a stand in a park during the summer months. He records the average weekly temperature and number of frozen fruit bars sold for 6 weeks. A 2-column table with 6 rows. The first column is labeled temperature (degrees Fahrenheit) with entries 67, 71, 76, 76, 82, 87. The second column is labeled fruit bars sold with entries 50, 54, 63, 65, 65, 100. What type of correlation exists between the temperature and the number of fruit bars sold? What is the real-world meaning of the slope of the line of best fit for the given scenario? There are approximately more fruit bars sold for every degree(s) the temperature rises.
The correlation that exists between the temperature and the number of fruits sold is positive but is almost negligible.
What is the correlation coefficient?The correlation coefficient helps us to know how strong is the relation between two variables. Its value is always between +1 to -1, where, the numerical value shows how strong is the relation between them and, the '+' or '-' sign shows whether the relationship is positive or negative.
1 indicates a strong positive relationship.-1 indicates a strong negative relationship.A result of zero indicates no relationship at all, therefore, independent variable.
We know that the correlation coefficient is given by the formula,
[tex]r =\dfrac{\sum\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sqrt{\sum\left(x_{i}-\bar{x}\right)^{2} \sum\left(y_{i}-\bar{y}\right)^{2}}}[/tex]
r = correlation coefficient
[tex]x_{i}[/tex] = values of the x-variable in a sample
[tex]\bar{x}[/tex] = mean of the values of the x-variable
[tex]y_{i}[/tex] = values of the y-variable in a sample
[tex]\bar{y}[/tex] = mean of the values of the y-variable
Calculate the value of [tex](x_i-\bar{x})(y_i-\bar{y}),\ (x_i-\bar{x})^2,{ \rm and}\ (y_i-\bar{y})^2[/tex] each separately, and then find the sum of them as shown below, therefore, the formula can be written as,
[tex]r =\dfrac{571.5}{261.5\times 1566.833333}= 0.0014[/tex]
Hence, the correlation that exists between the temperature and the number of fruits sold is positive but is almost negligible.
Learn more about Correlation Coefficients:
https://brainly.com/question/15353989
Answer:
1. positive
2. 2.2
3. 1
Step-by-step explanation:
There are 300 light bulbs lined up in a row in a long
room. Each bulb has its own switch and is currently
switched off. Each bulb is numbered consecutively from
1 to 270. You first flip every switch. You then flip the
switch on every second bulb(turning off 2, 4,6. ). You
then flip the switch on every third bulb (3, 6,. ). This
continues until you have gone through the process 300
times.
And how many of the light bulbs are illuminated now?
Flipping the bulbs is an illustration of sequence and patterns
17 bulbs would remain on.
How to determine the number of bulbs that will be on?The sequence of turning the bulbs off and on, would change the state of the bulbs until the 300th process.
While changing the state of the bulbs, the bulb at the perfect square positions would remain on.
There are 17 perfect square numbers between 1 and 300 (inclusive)
Hence, 17 bulbs would remain on.
Read more about sequence and patterns at:
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Please help me find the y-intercept and slope from these two-word problems. ANd tell me if they're proportional and negative or not.
Answer:
for the 1st one, m=10 and b=2 because from the word problem it says it starts at 10 which would be 10 on the y axis being where the line would pass through and spending 2 a week would be the slope because he is losing 2 per week its a negative slope so y=10x-2.
for the second one m=48 and b=44 its just like the last one but with different numbers but comment if you want me to explain it but the answer is y=48x-44.
15
The table below shows the lengths in
centimeters (cm) of a sample of 5 leaves
from a tree.
Leaf Length (cm)
14.2
2 13.8
12.6
13.4
5 11.5
WIN
3
4
?
A 6th leaf is added to the sample and its length is
measured. Its measure increases the mean value of
the sample of leaves but decreases the median value
of the sample of leaves. What is a possible
measurement for the length of the 6th leaf?
A) 13.1 cm
B) 13.3 cm
C) 13.4 cm
D) 13.7 cm
The mean measurement is the average value of the leaves
The possible measurement of the 6th leaf is 13.3
How to determine the possible value of the 6th sampleThe dataset is given as:
14.2, 13.8, 12.6, 13.4 and 11.5
Sort the data elements
11.5, 12.6, 13.4, 13.8 and 14.2
Calculate the mean
[tex]\bar x = \frac{11.5 + 12.6 + 13.4 + 13.8 + 14.2}{5}[/tex]
[tex]\bar x = \frac{65.5}{5}[/tex]
[tex]\bar x = 13.1[/tex]
So the mean is 13.1 and the median is 13.4
For the mean to increase, the 6th dataset must be greater than 13.1.
For the median to decrease, the 6th dataset must be less than 13.4
From the list of given options, the element that satisfy the above highlights is 13.3
Hence, the possible measurement of the 6th leaf is 13.3
Read more about mean and median at:
https://brainly.com/question/14532771
please help me pls ease
Answer:
8
Step-by-step explanation:
x = -2
the y to (-2,y) is 8
Calculate the monthly payment for a loan of $7,500 with an 11% interest rate compounded monthly over a period of 5 years.
Answer:
Option b - $163.07
Step-by-step explanation:
Given : A loan of $7,500 with an 11% interest rate compounded monthly over a period of 5 years.
To find : Monthly payment
Solution : Formula to find monthly payment
Monthly Payment = Amount / Discount factor
Discount factor
where ,
Amount = $7500
Rate= 11%=0.11
Time = 5 years
Now, put i and n in Discount factor ,
Monthly payment,
Therefore, Option b is correct monthly payment = $163.07
Answer:
The answer is $163.07
Step-by-step explanation:
To find the monthly payment:
[tex]i = \frac{0.11}{12} = 0.0091[/tex]
Time = 5 years
5 x 12 = 60
Discount factor
[tex]D = \frac{1-(1 + i) -^{n}}i[/tex]
[tex]D = \frac{1 - (1+ 0.009)-^{60} }{0.009} \\[/tex]
[tex]D = \frac{0.421}{0.009} = 46.77[/tex]
Monthly payment
[tex]\frac{7500}{46.77} = $163.07[/tex]
Which function BEST represents the graph?
Answer:
The last one
Step-by-step explanation:
K so first look at y intercept which is +5, so we eliminate A and C, then the graph opens up so the b in ax+bx+c has to be positive which is D
the point ( 0 ; 5 ) in on the curve
so f(0) = 5
choice between :
x² - 2x + 5 or x² + 2x + 5 then :
we have the point (1 ; 4)
with x² - 2x + 5
f(1) = 1² - 2*1 + 5 = 1 - 2 + 5 = 4
so it's correct
=> x² - 2x + 5 is the right one
I don't understand at all of how to do this
Answer:
17
Step-by-step explanation:
5x2=10
8 1/2x2=17
Hello Guys, I really need help with these questions. Here they are.
Solve each absolute value inequality and show its solution set.
(a) |x|<7
(b) |x+3|<9
(c) |y-8|>11
(g) |3x-1| greater or equal to 18
|4y+3| less than or equal to 13
Please help me with these. WIll give you 20 pts. I will award brainly for the best answer! Make the answers in numbers (inequalities)
Answer:
a.[tex]-7 < x < 7[/tex]
b.[tex]-12 < x < 6[/tex]
c. Two answers are: y < -3 or y > 19
d. Two answers are x ≤ -17/3 or x ≥ 19/3
e. -4 ≤ y ≤ 5/2
Step-by-step explanation:
Given the following questions:
Question one:
[tex]|x| < 7[/tex]
x is less than 7
[tex]=-7 < x < 7[/tex]
Question two:
[tex]|x+3| < 9[/tex]
[tex]x+3 > -9[/tex]
[tex]3-3=0[/tex]
[tex]-9+-3=-12[/tex]
[tex]x > -12[/tex]
[tex]x+3 < 9[/tex]
[tex]3-3=0[/tex]
[tex]9-3=6[/tex]
[tex]x < 6[/tex]
x is greater than -12, while less than 6
[tex]-12 < x < 6[/tex]
Question three:
[tex]|y-8| > 11[/tex]
[tex]y-8 < -11[/tex]
[tex]-8+8=0[/tex]
[tex]-11+8=-3[/tex]
[tex]y < -3[/tex]
[tex]y-8 > 11[/tex]
[tex]-8+8=0[/tex]
[tex]11+8=19[/tex]
[tex]y > 19[/tex]
Two answers are: y < -3 or y > 19
Question four:
|3x - 1| ≥ 18
3x - 1 ≤ -18
-1 + 1 = 0
-18 + 1 = -17
3x ≤ -17
x ≤ -17/3
3x - 1 ≥ 18
-1 + 1 = 0
18 + 1 = 19
3x ≥ 19
x ≥ 19/3
Two answers are: x ≤ -17/3 or x ≥ 19/3
Question five:
|4y + 3| ≤ 13
|4y + 3| ≤ 13 = -13 ≤ 4y + 3 ≤ 13
-13 ≤ 4y + 3 ≤ 13
4y + 3 ≥ -13
3 - 3 = 0
-13 + - 3 = -16
4y ≥ -16
4 ÷ - 16 = -4
y ≥ -4
4y + 3 ≤ 13
3 + -3 = 0
13 - 3 = 10
4y ≤ 10 = 10/4 ÷ 2 = 5/2
y ≤ 5/2
-4 ≤ y ≤ 5/2
Hope this helps.
Two angles are complementary. One angle measures 28 What is the measure of the other angle?
the sum of two complementary angles is 90°
so 28 + ? = 90
? = 62°
here'
complementry angle means those whose sum is 90
here one angle is given so,
another angle =90-28
that is 72.ans
i need help with this equation its confusing ASAP
Answer:
multiply both sides by 3
Step-by-step explanation:
i would multiply 3 on both sides to cancel the fraction out so i dont have to work with fractions
Answer:
multiply both 3 sides
Step-by-step explanation:
real simple m
solve the simultanious equasion
8x+3y=45
2x+3y=27
pls pls pls help
Answer:
[tex]x=3,\:y=7[/tex]
Step-by-step explanation:
[tex]\begin{bmatrix}8x+3y=45\\ 2x+3y=27\end{bmatrix}[/tex]
[tex]\mathrm{Substitute\:}x=\frac{45-3y}{8}[/tex]
[tex]\begin{bmatrix}2\cdot \frac{45-3y}{8}+3y=27\end{bmatrix}[/tex]
[tex]\begin{bmatrix}\frac{45+9y}{4}=27\end{bmatrix}[/tex]
[tex]y=7[/tex]
[tex]\mathrm{For\:}x=\frac{45-3y}{8}[/tex]
[tex]\mathrm{Substitute\:}y=7[/tex]
[tex]x=\frac{45-3\cdot \:7}{8}[/tex]
[tex]x=3[/tex]
[tex]\mathrm{The\:solutions\:are}:[/tex]
[tex]x=3,\:y=7[/tex]
Two angles are supplementary. One angle is 36 degrees bigger than the other. What is the measure of the bigger angle
let the other angle be x degree
bigger angle = x +36°
According to the question,
x+36° + x° = 180°
=> 2x = 180–36
=> x = 144/2
=> x = 72°
Measure of the bigger angle = x + 36
= 72° + 36°
= 108°
what is the answer???????????
Answer:
Step-by-step explanation:
First we make the denominators the same:
2/3 = 10/15
2/5 = 6/15
Ratio: 6/15 : 10/15
Have a great day!!
Please rate and mark brainliest!!
532^7•532^-8•532^3
How do I do this it’s hard
Answer:
283024
Step-by-step explanation:
First do 532^7
The answer is 12060947473224100000
Next, do 532^-8
The answer is 1/6416424055755230000000
Multiply that number by the first number
(I don't want to type that all out again)
The answer is 1/532
Now, do the last problem. 532^3
The answer is 150568768
Multiply that number by 1/532
The answer is 283024
Hope I helped :)
Please consider Brainliest :)
(it took me so long to type that)
Part D In the scale drawing, what will the length of the west side of the backyard be?
NEED HELP ASAP WILL GIVE BRAINIST TO BEST AWNSER
Answer: answer witten in the picture attached to this......
f(x)=1/3x-4 is it linear or exponential?
Answer:
Linear
Step-by-step explanation:
since the x value has no exponent to it, and is simply on its own, it is linear
If the ratio of a:b is 4:15 and the ratio of b:c is 5:9, what is the ratio of a:c?
Answer:
Your answer should be 4:9
Step-by-step explanation:
Good luck.
When simplifying expressions that involve many exponent properties, how do you decide what to do first?
first (...), after √ and squares from left to right then from left to right . and : and lastly + and -
HELP i need help quickly please
Answer:
c
Step-by-step explanation:
For which values of c does the quadratic equation x^2-2x+c=0 Two roots of opposite signs
Answer:
-2
Step-by-step explanation:
The value of c is -4, therefore the additive inverse should be added to get the answer 0
The roots of the quadratic equation x² – 2x + c = 0 are a and –b. Then the value of c is a² – 2a.
What is a quadratic equation?It is a polynomial that is equal to zero. Polynomial of variable power 2, 1, and 0 terms are there. Any equation having one term in which the power of the variable is a maximum of 2 then it is called a quadratic equation.
For which values of c does the quadratic equation
x² – 2x + c = 0 ...1
Two roots of opposite signs
Let the two roots of the opposite signs be a and –b
x = a, –b
Then we have factorized the equation,
(x + b)(x – a) = 0
On solving, we have
x² + (b – a)x – ab = 0 ...2
In comparing equation 2 with equation 1, we have
⇒ b – a = –2
⇒ b = –2 + a
and
⇒ c = a(a – 2) = a² – 2a
More about the quadratic equation link is given below.
https://brainly.com/question/2263981
What is the value of x?
x=32 degrees
Step-by-step explanation:
Angles in equilateral triangles are equal, each angle is 60 degrees
2x-4 =60
+4
2x = 64
÷2
x = 32
Also, y will be 12 (12x5=60)
Hope this helps!
what is the answer for thiss please helpp
3-2x<1
Answer:
x > 1
Step-by-step explanation:
3 - 2x < 1
- 2x < 1 - 3
- 2x < - 2
Divide - 2 on both sides,
- 2x / - 2 > - 2 / - 2
x > 1
Answer:
x>1
Step-by-step explanation:
3-2x<1-2x<1-3-2x<-2x>-2/-2[Sign alter because dividing by negative number]Hence, x>1
What is the equation written in vertex form of a parabola with a vertex of (–1, 8) that passes through (1, 0)?
Answer:
y = -2(x + 1)^2 + 8
Step-by-step explanation:
The equation of a parabola can be written in the form;
y = a(x-h)^2 + k
where a is the multiplier (h,k) is the vertex
so h = -1 and k = 8
Plug in these values
y = a(x + 1)^2 + 8
So to get the value of a, we use the point where the parabola passes through which is the point (1,0)
Simply substitute the values of x and y
0 = a(1 + 1)^2 + 8
0 = a(2)^2 + 8
-8 = 4a
a = -8/4
a = -2
So therefore the equation of the parabola is ;
y = -2(x + 1)^2 + 8
The required equation of parabola with given vertex and point of passing is [tex]y=-2(x+1)^{2}+8[/tex].
Given data:
The vertex of parabola is, (-1, 8).
The vertex is passing through (1, 0).
We know that standard equation parabola is,
[tex]y=a(x-h)^{2}+k[/tex]
Here, a is any multiplier, h and k are the vertices.
From the given problem,
h = -1
k = 8
So substituting the values as,
[tex]y=a(x-(-1))^{2}+8\\y=a(x+1)^{2}+8[/tex]..............................................(1)
To obtain the value of a, we use the point where the parabola passes through which is the point (1,0). Here, x = 1 and y =0.
So,
[tex]0=a(1+1)^{2}+8\\\\a = -2[/tex]
Substitute the value of a in equation (1) as,
[tex]y=(-2)(x+1)^{2}+8\\\\y=-2(x+1)^{2}+8[/tex]
Thus, the required equation of parabola is [tex]y=-2(x+1)^{2}+8[/tex].
learn more about the parabola here:
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What is the additive inverse of the number 1.6?
A. -1.6
B. 0
C. 1
D. 6.1
Answer:
-1.6
Step-by-step explanation:
Because the additive inverse of any positive number is it's negative number
Foster is centering a photo that is 9 1/2 inches wide on a scrapbook page that is 14 inches wide. How far from each side of the page should he put the picture? Enter your answer as a mixed number.
Step-by-step explanation:
Given that he dimension is
9 1/2 inches wide
and 14 inches wide.
half way from 9.5 inches is
4.75in
half way from 14 inches is
7in
Therefore he put the picture at 4.75in from the top and 7in from the sides
If f(x) = 5x +5, which of the following is the inverse of f(x)?
=
=
(x)
3
(x+)
O A. F'(x) = 5*55
-(
O B. F'(x) = 36445)
f-
O c. f'(x) = 5(x+5)
-=
O D. +++(x) = 34,553
f-
5
3
(-)
Answer: D
Step-by-step explanation:
Three straight lines are shown in the diagram. Work out the sizes of angles a,b and c.
15 points to whoever gets this by the way.
Answer:
b = 100°
should be the answer