The first step in finding the given operations for the functions is to substitute the expressions for f(x) and g(x) into the operations.
For the first set of functions, f(x) = x2 and g(x) = 4x − 3, we have:
(f + g)(x) = f(x) + g(x) = x2 + (4x - 3) = x2 + 4x - 3
(f - g)(x) = f(x) - g(x) = x2 - (4x - 3) = x2 - 4x + 3
(fg)(x) = f(x)g(x) = x2(4x - 3) = 4x3 - 3x2
(f/g)(x) = f(x)/g(x) = x2/(4x - 3)
For the second set of functions, f(x) = x2 + 5 and g(x) = 7 - x, we have:
(f + g)(x) = f(x) + g(x) = x2 + 5 + (7 - x) = x2 - x + 12
(f - g)(x) = f(x) - g(x) = x2 + 5 - (7 - x) = x2 + x - 2
(fg)(x) = f(x)g(x) = (x2 + 5)(7 - x) = 7x2 - x3 + 35 - 5x
(f/g)(x) = f(x)/g(x) = (x2 + 5)/(7 - x)
The given operations are:
1.) (f + g)(x) = x2 + 4x - 3, (f - g)(x) = x2 - 4x + 3, (fg)(x) = 4x3 - 3x2, (f/g)(x) = x2/(4x - 3)
2.) (f + g)(x) = x2 - x + 12, (f - g)(x) = x2 + x - 2, (fg)(x) = 7x2 - x3 + 35 - 5x, (f/g)(x) = (x2 + 5)/(7 - x)
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Guide
The table shows the height of a child, in contimeters, at different ages.
Height of a Child
Age (years)
2
3
5
7
15
12
Holght (cm)
84.5
90.5
105.0
119.0
125.5
151.5
Create an equation to model the relationship between the height, in centimeters, and the age, in years, of this child.
Show your work or explain your equation. Be sure to include the limitations of your equation.
Enter your equation, your work or explanation, and the limitations in the box provided.
The estimated height of the child at delivery is represented by the y-intercept, which is 72.66.
What is an equation to model the relationship between the height, in centimeters, and the age, in years, of this child?We may use linear regression to construct an equation that represents the link between this child's height and age. The line of best fit for the supplied data is determined by linear regression, which can be written as:
Age (years) + b * m * Height (cm)
where m denotes the line's slope and b its y-intercept.
The equation that most closely matches the above data can be determined using a calculator or spreadsheet software as follows:
Age (years) + 72.66 Height (cm) = 6.087
Thus, the height of the youngster increases by roughly 6.087 centimetres for every year that their age rises. The estimated height of the infant at delivery is represented by the y-intercept, which is 72.66.
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Felipe makes a bracelet with 30 beads. The table shows the number of beads of
each color on the bracelet. Write 3 ratios to represent the number of beads (part-
part, part-part, and part-whole)
Other fractions could represent the part of the beads on the bracelet that will be green are : 1/3, 2/6, 3/9, 4/12.
What is fraction?A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.
here, we have,
The maximum number of beads the bracelet can have = 12 beads
The fraction of green beads in the bracelet = 1/3
⇒ 1 bead in every 3 bead is GREEN.
⇒The minimum number of total beads = Number of (green+ Other) bead = 2 + 1 = 3
Now, the possible number of total beads the bracelet can have is
3, 4, 5 , 6, 7, 8, 9 ,10, 11 and 12.
If there are total 3 beads, the fraction representing the green bead = 1/3
If there are total 6( =3 +3) beads, the fraction representing the green bead
= 1 green bead in first 3 beads + 1 green bead in next 3 beads
= 2 green beads in total 6 beads
= 2/6
If there are total 9( = 3 +3 +3) beads, the fraction representing the green bead = 3/9
If there are total 12( = 3 + 3 +3 +3) beads, the fraction representing the green bead = 4/12
Hence, other fractions could represent the part of the beads on the bracelet that will be green are :1/3, 2/6, 3/9, 4/12.
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Pippin has a 32-ounce sports drink. She drinks 20 ounces.
Enter the percentage of ounces Pippin has left of her sports drink. HELP PLSS
THIS IS DUE TONIGHT AND IT COUNTS TOWARDS MY GRADE
Pippin has 37.5% of her sports drink left.
What is Percentage?
Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. Percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%".
Pippin has 32 - 20 = 12 ounces of sports drink left.
To find the percentage of ounces Pippin has left, you need to divide the number of ounces left by the original amount of sports drink (32) and then multiply by 100 to get the percentage:
(12/32) x 100 = 37.5%
Therefore, Pippin has 37.5% of her sports drink left.
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Use Euclid's first book to prove what specific quadrilaterals are produced by non-perpendicular, unequal, non-bisecting diagonals.
Using the definition of a parallelogram in Euclid's first book, we can say that a non-perpendicular, unequal, non-bisecting diagonal produces either a trapezium or a rhombus if the opposite angles are equal, then it is a rhombus, and if the opposite angles are unequal, then it is a trapezium.
To use Euclid's first book to prove what specific quadrilaterals are produced by non-perpendicular, unequal, non-bisecting diagonals, start by considering Euclid's definitions of a parallelogram, a trapezium, and a rhombus.
In Euclid's first book, a parallelogram is defined as a quadrilateral which has two pairs of equal and parallel sides, while a trapezium is defined as a quadrilateral which has two pairs of opposite sides parallel, but with the other two sides unequal and non-parallel.
Finally, a rhombus is defined as a quadrilateral (parallelogram) which has all four sides equal, and with the opposite angles equal.
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Alexandra buys a swing set online for $745.If shipping and handling are an additional 21% of the price,how much shipping and handling will Alexandra pay?
Answer:
$156.45
Step-by-step explanation:
Swing set $745
21/100 x $745 = 156.45
Please be meeeeee!!!
Answer:
answer is sus.
Step-by-step explanation:
sus is the correct answer to your problrm. hope this helps!
Select the correct answer.
What is the end behavior of this radical function?
f(x) = 3√x-4
As x approaches positive infinity, f(x) approaches positive infinity.
As x approaches negative infinity, f(x) approaches positive infinity.
As x approaches positive infinity, f(x) approaches negative infinity.
As x approaches negative infinity, f(x) approaches 0.
Answer:
The correct answers are:
As x approaches positive infinity, f(x) approaches positive infinity.
As x approaches negative infinity, f(x) approaches 0.
Step-by-step explanation:
please help I have new clue what it is asking.
Answer:
110° and 215°
Step-by-step explanation:
the bearing of one point to another is the measure of the clockwise angle from the north line N at the point C to the point D , that is
(a)
bearing of D from C is 110° ( purple shaded angle )
(b)
the bearing of D from C is 215° ( blue shaded angle )
7x - 4y = -8
X + 2y = -14
Step-by-step explanation:
First of all sorry for the bad hand writing
so rn I'm using elimination method which I try to eliminate y by multiple number 2 equation with 2
in order to find x
I know my explanation is quite confusing but any way goodluck n hope u understand
Heatrow airport in london england employs 80 chefs to prepare meals for international flight
Please show your work.
We can write the perimeter of the equilateral triangle as -
P = 13x - 19.
What is line of best fit?A line of best fit is a straight line that minimizes the distance between it and some data. The line of best fit is used to express a relationship in a scatter plot of different data points. It is an output of regression analysis and can be used as a prediction tool for indicators and price movements.Given is a equilateral triangle.
We can write the perimeter of a equilateral triangle as -
Perimeter = (2x + 3) + (4x - 5) + (7x - 17)
Perimeter = 2x + 4x + 7x + 3 - 5 - 17
Perimeter = 13x - 19
Therefore, we can write the perimeter of the equilateral triangle as -
P = 13x - 19.
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Write the given expression as a single trigonometric function. tan(3pi/7) - tan(2pi/5)/ 1+tan(3pi/7) tan(2pi/5)
On sοlving the given trigοnοmetric expressiοn using the trigοnοmetric identity, we fοund that it is equal tο tan (π/35).
What is an expressiοn?Mathematical expressiοns cοnsist οf at least twο numbers οr variables, at least οne maths οperatiοn, and a statement. It's pοssible tο multiply, divide, add, οr subtract with this mathematical οperatiοn. A finite cοmbinatiοn οf symbοls that are well-fοrmed in accοrdance with cοntext-dependent principles is referred tο as an expressiοn οr mathematical expressiοn.
There are twο sοrts οf expressiοns in mathematics: numerical expressiοns, which οnly cοntain numbers, and algebraic expressiοns, which alsο include variables. Expressiοns can be categοrised as arithmetic/numerical expressiοns, fractiοnal expressiοns, οr algebraic expressiοns depending οn the terms they cοntain.
The given trigοnοmetric expressiοn is:
[tex]\frac{tan (3\pi /7) - tan (2\pi /5)}{1+tan (3\pi /7) tan (2\pi /5)}[/tex]
tan (a-b) = sin(a-b) / cοs (a-b)
= (sin a cοs b - cοs a sin b) / (cοs a cοs b + sin a sin b)
Dividing the numeratοr and denοminatοr by (cοs a cοs b), we get,
tan (a-b) = (tan (a) - tan (b)) / (1+tan a tan b)
This expressiοn is similar tο the given expressiοn.
Cοmparing,
a = 3π/7
b = 2π/5
[tex]\frac{tan (3\pi /7) - tan (2\pi /5)}{1+tan (3\pi /7) tan (2\pi /5)}[/tex] = tan( 3π/7 - 2π/5) = tan (π/35)
Therefοre on solving the given trigonοmetric expression using the trigonοmetric identity, we found that it is equal tο tan (π/35).
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Help please it would mean a lot
The number of pounds of peaches sold is given as follows:
65 pounds, option 2.
How to obtain the amount?The amount is obtained solving a system of equations, for which the variables are given as follows:
Variable x: pounds of apples.Variable y: pounds of peaches.She sold a total of 165 pounds, hence:
x + y = 165
x = 165 - y.
She made $337.50, hence:
1.75x + 2.5y = 337.50.
Replacing the first equation into the second, the value of y is obtained as follows:
1.75(165 - y) + 2.5y = 337.50
0.75y = 48.75
y = 48.75/0.75
y = 65 pounds.
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Does anyone know how to solve this
y=-2x^(2) + 12x - 15 ;x=3
Cause when I try it I get y=-27
but answer is supposed to be y=3 and x=3
Pls explain how
Answer: Order of operations
Step-by-step explanation:
Plug in 3 for x so...
y = -2(3)² + 12(3) - 15
Don't forget the order of operations, exponents go first
y = -2(9) + 12(3) - 15
Then multiply
y = -18 + 36 - 15
Simplify
y = 3
Suppose that a, b and c are non-zero real numbers. Determine whether or not it is possible for each of the six quadratic equations
ax2 + bx + c = 0
ax2 + cx + b = 0
bx2 + ax + c = 0
bx2 + cx + a = 0
cx2 + ax + b = 0
cx2 + bx + a = 0
to have at least one real root.
It is possible for each of the six quadratic equations to have at least one real root.
This is because the discriminant of a quadratic equation determines the nature of its roots. The discriminant is given by the formula: D = b² - 4ac.
If D > 0, the equation has two distinct real roots.
If D = 0, the equation has one real root (a double root).
If D < 0, the equation has no real roots (two complex roots).
In each of the six equations, the coefficients a, b, and c are non-zero real numbers. This means that the discriminant can be positive, zero, or negative, depending on the values of a, b, and c. Therefore, it is possible for each of the six equations to have at least one real root.
For example, consider the equation ax² + bx + c = 0. If b² > 4ac, then the equation has two distinct real roots. If b² = 4ac, then the equation has one real root (a double root). If b² < 4ac, then the equation has no real roots (two complex roots).
Similarly, the other five equations can also have at least one real root, depending on the values of the coefficients a, b, and c.
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Evaluate the function when x = 3.
f(x) = 5x + 2
A. -1/5
B. 1
C. 3
D. 7
Answer:
To evaluate the function when x = 3, we substitute 3 for x in the given function f(x) = 5x + 2:
f(3) = 5(3) + 2
f(3) = 15 + 2
f(3) = 17
Therefore, when x = 3, the value of the function f(x) = 5x + 2 is 17.
So, the correct answer is option D, 7 is incorrect because 5*3+2 = 17.
The Answers given are incorrect this is because the correct answer is 5.7
f(x) = 5x + 2
f(3)=5(3)+2
3f=15+2
3f=17
3f÷3=17÷3
f=5.66666666667
=5.7
. For each situation below, state the transformation(s) that occur between
f(x) and g(x).
(please fill out the whole thing I need it by tomorrow)
The transformations that are occurring between each of the function are given as follows:
f(x) = 14(x - 3) + 1 and g(x) = 7(x + 2) + 1: vertical compression by a factor of 1/2 and shift left 5 units.
f(x) = 6(x + 2) - 3 and g(x) = 8(x - 3) + 6: vertical stretch by a factor of 4/3, shift right 5 units and shift up 9 units.
f(x) = 3(x - 1) + 2/3 and g(x) = 3(3 - x) - 2/3: reflection over the y-axis, shift left 2 units and shift up 4/3 units.
f(x) = 1/3(x + 2) - 4 and g(x) = 1/3(-x - 2) + 6: reflection over the y-axis and shift up 10 units.
f(x) = 4/3(x - 2/3) + 1/3 and g(x) = 2/3(2/3 - x) = reflection over the y-axis, shift down of 1/3 units and vertical stretch by a factor of 2.
What are transformations?The curve of the graph "moves to the left/right/up/down," "it expands or compresses," or "it reflects" when a function is changed.
The transformation rules are defined as follows:
x -> x + a: shift left a unit.
x -> x - a: shift right a unit.
y -> y + a: shift up a unit.
y -> y - a: shift down a unit.
y -> ky, k > 1: vertical stretching by k times.
Vertical compression by a factor of k is achieved by y -> ky, k 1, horizontal compression by a factor of k is achieved by x -> kx, k > 1, and vertical stretch by a factor of k is achieved by x -> kx, k 1.
x -> -x: reflection over the y-axis.
y -> -y: reflection over the x-axis.
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ins all the integers from -7 through 4 , inclusive. Set Q contains the absolute values of all the numbers in Set P. numbers are in the intersection of sets P and Q ?
The numbers in the intersection of sets P and Q are 0, 1, 2, 3, and 4.
The integers in Set P are {-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4}.
The absolute values of these integers are {7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4}, which make up Set Q. The intersection of Set P and Set Q are the integers that are in both sets, which are {0, 1, 2, 3, 4}.
Therefore, the numbers in the intersection of sets P and Q are 0, 1, 2, 3, and 4.
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Alexia took out a 12-year loan for $72,000 to renovate her home. If her monthly payments are $680, what is the interest rate?
Answer:
Finding the total accrued amount is necessary before we can determine the interest rate.
Simply multiplying the monthly payment by the number of months in a 12-year period will give us the accrued amount.
144 months in 12 years.
$680 x Monthly Payment
A = 680 * 144
A = $97920
We can use the following formula now that we know the total accrued sum:
A= 97920
P = 72000
t= 12
Let's replace them with our values now.
The percentage value is then obtained by multiplying the r value by 100.
0.03 x 100 = 3%
The interest rate for Alexia is 3%.
How many solutions exist for 1/1-x^2=-|3x-2|+5
A. 1
B. 2
C. 3
D. 4
23 POINTS!
Answer:
4 !!!!
Step-by-step explanation:
f (x) = 1/1-x2 g (x) = -(3 x -2 ) - 5 then the number of solutions if the number of intersections of f(x) and g (x) i drew a graph and there was 4 intersections so thats how i got this answer
I hope i was right
There are two solutions to the equation 1/(1 - x²) = -|3x - 2| + 5.
What is Equation?An equation is a statement that shows that two mathematical expressions are equal to each other. It typically consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, and exponents.
We can start by considering the case where the expression inside the absolute value is non-negative, i.e., 3x - 2 ≥ 0, which implies x ≥ 2/3. In this case, the equation becomes:
1/(1 - x²) = 3x - 2 + 5
1/(1 - x) = 3x + 3
Multiplying both sides by 1 - x², we get:
1 = (3x + 3)(1 - x²)
3x³ + x² - 3x - 2 = 0
This equation has one real root.
The root is approximately x = 0.727.
Next, we consider the case where the expression inside the absolute value is negative, i.e., 3x - 2 < 0, which implies x < 2/3. In this case, the equation becomes:
1/(1 - x²) = -(3x - 2) + 5
Simplifying this equation, we get:
1/(1 - x) = -3x + 3
1 = (-3x + 3)(1 - x²)
3x³ - x² - 3x + 2 = 0
This equation also has one real root, which we can find using the same methods as before. The root is approximately x = -0.727. \
Therefore, there are two solutions to the equation 1/(1 - x²) = -|3x - 2| + 5.
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Admission to the museum is $8. Students receive a 30% discount. How much de
students pay? How much is the discount worth?
Answer:
The students would be paying a low amount of $5.60. The 30% discount takes off $2.40 of the original price, so the discount is worth $2.40
Solve the syslem by either the addition mothod or the substitution method. y=2x-7 y=4x-21
Using the addition method the solution to the system is (x,y) = (14/3, 7/3) and using the substitution method the solution to the system is (x,y) = (7,7).
Using the addition method, we can rearrange the equations so that one of the variables cancels out:
y=2x-7 -> -2x + y = -7
y=4x-21 -> -4x + y = -21
Now, we can add the two equations together to eliminate the y variable:
-2x + y = -7
-4x + y = -21
______________
-6x = -28
Next, we can solve for x:
x = 28/6 = 14/3
Now, we can substitute this value of x back into one of the original equations to find y:
y = 2x - 7 = 2(14/3) - 7 = 28/3 - 21/3 = 7/3
So the solution to the system is (x,y) = (14/3, 7/3).
Using the substitution method, we can substitute one equation into the other to eliminate one of the variables. For example, we can substitute the first equation, y=2x-7, into the second equation, y=4x-21:
2x-7 = 4x-21
Next, we can rearrange the equation to solve for x:
2x = 14
x = 7
Now, we can substitute this value of x back into one of the original equations to find y:
y = 2x - 7 = 2(7) - 7 = 7
So the solution to the system is (x,y) = (7,7).
In conclusion, we can solve the system y=2x-7 and y=4x-21 using either the addition method or the substitution method to find the solution (x,y).
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Where does the FDIC’s reserve fund come from?
a.
The FDIC has access to federal tax revenue.
b.
If an insured bank fails, the FDIC keeps the money at that bank that is beyond the insured limit.
c.
A certain amount of money goes directly from the Treasury to the FDIC’s reserve fund.
d.
Insured banks pay a premium on the money insured.
Answer:
d. Insured banks pay a premium on the money insured. The FDIC's reserve fund comes from premiums paid by insured banks. The premiums are paid based on the amount of deposits held by the bank and the level of risk the bank poses to the FDIC's insurance fund.
Solve the following rational equation: If no solution exists then state that. Show complete work. y/y−2y+ 4/y-3 = 4/ y2−5y+6
The answer: no real solutions to this equation
To solve the equation, we need to find a common denominator for all of the fractions on the left side of the equation. The common denominator will be (y-2)(y-3), so we will multiply each fraction by the appropriate factor to get the common denominator.
[tex]y/y -2y+ 4/y-3 = 4/ y2-5y+6[/tex]
[tex]y(y-3)/(y-2)(y-3) - 2y(y-2)/(y-2)(y-3) + 4(y-2)/(y-2)(y-3) = 4/(y-2)(y-3)[/tex]
Now we can combine the fractions on the left side of the equation:
[tex](y2-3y-2y2+4y+4y-8)/(y-2)(y-3) = 4/(y-2)(y-3)[/tex]
Simplifying the numerator on the left side gives us:
[tex](-y2+5y-8)/(y-2)(y-3) = 4/(y-2)(y-3)[/tex]
Now we can cross-multiply and simplify:
[tex](-y2+5y-8) = 4[/tex]
[tex]-y2+5y-12 = 0[/tex]
To solve this equation, we can use the quadratic formula:
[tex]y=\left(-5\pm \sqrt{52-4\left(-1\right)\left(-12\right)}\right)[/tex]
[tex]y=\left(-5\pm \sqrt{25-48}\right)[/tex]
[tex]y=\left(-5\pm \sqrt{-23}\right)[/tex]
Since the square root of a negative number is not a real number, there are no real solutions to this equation. Therefore, the answer is no solution.
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"Find the total price: Total bill: $45.65 Tip: 22% Tax: 5%"
Answer:
To find the total price, we need to add the tip and tax to the total bill amount.
First, we need to find the amount of the tip. To do this, we will multiply the total bill by 22% (or 0.22):
$45.65 x 0.22 = $10.04
This means the tip amount is $10.04.
Next, we need to find the amount of tax. To do this, we will multiply the total bill by 5% (or 0.05):
$45.65 x 0.05 = $2.28
This means the tax amount is $2.28.
Finally, we can find the total price by adding the total bill, tip, and tax together:
$45.65 + $10.04 + $2.28 = $57.97
Therefore, the total price is $57.97.
Dylan has two measuring jugs, A and B, which hold a combined volume of 800ml. Dulan fill jug A with water. He then fills jug B by pouring in water from jug A. After this, there is 200 ml of water remaining in jug A. Work out the volune of each measuring jug
Jug A has a vοlume οf 400 ml and jug B has a vοlume οf 400 ml.
What is the equatiοn in twο variables?An equatiοn in twο variables is an equatiοn that invοlves twο variables, typically x and y, and relates them with cοnstants and mathematical οperatiοns such as additiοn, subtractiοn, multiplicatiοn, and divisiοn.
Let's represent the vοlume οf jug A by x and the vοlume οf jug B by y.
We knοw that the cοmbined vοlume οf the twο jugs is 800 ml:
x + y = 800 (equatiοn 1)
We alsο knοw that after pοuring water frοm jug A tο jug B, there is 200 ml οf water remaining in jug A. This means that jug B cοntains x ml οf water:
y = x (equatiοn 2)
We can substitute equatiοn 2 intο equatiοn 1 tο eliminate y:
x + y = 800
x + x = 800
2x = 800
x = 400
Nοw that we knοw that the vοlume οf jug A is 400 ml, we can use equatiοn 2 tο find the vοlume οf jug B:
y = x
y = 400
Therefοre, the vοlume οf jug B is alsο 400 ml.
Hence, jug A has a vοlume οf 400 ml and jug B has a vοlume οf 400 ml.
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Please answer this question
Answer:
14
Step-by-step explanation:
Help me fast its late and I gotta sleep this was due 6 minutes ago, I barely understand anything from this lesson and I don't got the best teacher so please help
Answer:
Step-by-step explanation:
Im not sure I can't see the picture.
Answer: Hence, the width of the path is 4 meters.
First we should find the area of the pool. Our width is 14 and our length is 18. In order to find the area, we must multiply these two numbers. 14 times 18 is 252. This means that the area of the pool is 252.
Since we know that one side of the pool is 18 and one side is 14, we can write out the equation below.
572= (18+2x)(14+2x)
The first thing we should do is combine like-terms.Our equation would now be as below.
x^2+16-80=0
Next we would take our quadratic equation- x=-b+-√b2-4ac/2a and fill it in with the numbers that we had in the first equation, our equation would be like below. A=1 B=16 C=-80
x=-16+-√16^2-4* 1 * -80 /2 * 1
When then simplify our equation leaving us with x=-16+-√16^2-4*1(-80)/2 * 1, which broken down is x=-16+-√+-24/2.
Lastly we must subtract and add to receive our two answers.
One side is 4 and the other is -20. Since the width of the walk cannot not be a negative number, we know that the width of the path is 4 meters.
Hence, the width of the path is 4 meters.
I hope this helped & Good Luck <3!!!
PLSSSS HELP IF YOU TURLY KNOW THISSS
Answer:
x = -6
Step-by-step explanation:
-4 + 2x = 4x+8
-4-8 = 4x - 2x
-12 = 2x
x = -6
Answer:
x = - 6
Step-by-step explanation:
Hope this helps! <33
Question 16. Let A be an m×n matrix, X an n×r matrix, and B an m×r matrix. Show that AX =B if and only if Axj=bj, j=1,…,r
Axj = bj, j=1,…,r.
To show that AX = B if and only if Axj = bj, j=1,…,r, we can start by recognizing that AX is an m×r matrix and B is an m×r matrix. Since the dimensions of these two matrices are the same, they are equal if and only if each corresponding element is equal.
Therefore, we can conclude that AX = B if and only if each corresponding element of AX is equal to its corresponding element in B. Mathematically, this is expressed as Axj = bj, j=1,…,r.
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