Let f (x) be a quadratic equation of the form:
[tex]x^2+bx+c\\[/tex]
Where b and c are real numbers.
So a generic procedure for completing squares is:
1) identify b
In this case b = 10
2) divide b between 2 and then square it
[tex](\frac{b}{2})^ 2\\[/tex]
[tex](\frac{10}{2})^ 2=25\\[/tex]
3) Add and subtract [tex]b ^ 2[/tex] in the equation
[tex]x ^2+10x + 25 - 25 = 50\\[/tex]
[tex]x ^ 2 + 10x +25 = 75\\[/tex]
4) Rewrite the equation as follows
[tex](x+\frac{b}{2})^2=c\\[/tex]
[tex](x + 5) ^ 2 = 75\\[/tex]
The number that you must add to both sides of the equation is number [tex]25 = (\frac{b}{2})^2[/tex]
The completed form of the equation is (x + 5)² = 75.
To solve the equation x² + 10x = 50 by completing the square, you need to add a specific number to both sides to create a perfect square trinomial on the left side. Here’s how you do it step-by-step:
Start with the equation: x² + 10x = 50.Take the coefficient of the x term (which is 10), divide it by 2, and then square the result: (10/2)² = 25.Add 25 to both sides of the equation: x²+ 10x + 25 = 50 + 25.Now the left-hand side of the equation can be factored as a perfect square trinomial: (x + 5)² = 75.Therefore, the number that must be added to both sides to complete the square is 25.
Which platonic solid has twelve faces that are regular pentagons
Answer:
Dodecahedron
Step-by-step explanation:
Maria had planned to walk 10 1/8 miles on wednesday, if she walked 7 2/8 miles in the morning how far would she need to walk in tge afternoon ?
Answer:
Distance Maria had to walk in the afternoon is [tex]2\frac{7}{8}[/tex]
Step-by-step explanation:
Lets take that Maria walked [tex]x[/tex] miles in the afternoon.
Distance walked in the morning + distance walked in the afternoon = [tex]10\frac{1}{8}[/tex]
⇒[tex]7\frac{2}{8} +x=10\frac{1}{8}[/tex]
[tex]\frac{58}{8} +\frac{8x}{8} =\frac{81}{8}[/tex]
[tex]\frac{58+8x}{8} =\frac{81}{8}[/tex]
[tex]58+8x=81[/tex]
[tex]8x=81-58\\[/tex]
[tex]8x=23[/tex]
[tex]x=\frac{23}{8}[/tex]
[tex]x=2\frac{7}{8}[/tex]
Which expression is the completely factored form of 27x^3+y^6
Answer:
your answers C
Step-by-step explanation:
The factored form of the expression [tex]27x^3+y^6[/tex] is [tex](3x+y^2)(9x2-3xy^2+y^4)[/tex]. The correct answer of the factored expression is option A
Given the expression;
[tex]27x^3+y^6[/tex]
This can be expressed as the sum of cubes
[tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]
The given expression can also be written as the sum of cubes as shown:
[tex]27x^3+y^6 = (3x)^3+(y^2)^3[/tex]
We can see that:
a = 3x
b = y²
Substituting the given parameters into the formula above;
[tex]a^3+b^3=(a+b)(a^2-ab+b^2)\\(3x)^3+(y^2)^3=(3x+y^2)((3x)^2-3xy^2+(y^2)^2)\\(3x)^3+(y^2)^3=(3x+y^2)(9x2-3xy^2+y^4)\\[/tex]
Hence the factored form of the expression is [tex](3x+y^2)(9x2-3xy^2+y^4)[/tex]
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Wher whould -0.5 and -1 and -1/4 and 0 go on a number line
What is the remainder when the polynomial 5x2+10x−15 is divided by x + 5? Enter your answer in the box.
There are two ways you can approach this:
1) Factor [tex]\frac{5(x-1)(x+3)}{x+5}[/tex] this equals to [tex]\frac{5x^2+10x-15}{x+5}[/tex]
2) or divide which equals to 5x+35+[tex]\frac{160}{x+5}[/tex]
Doughnuts are sold in bags anf cartons a bag holds 4 doughnuts and a carton holds ten doughnuts. Tom buys b bags of dougnuts and c cartons of doughnuts he buys a total of t doughnuts. Erite down a formula for t in terms of b and c
The formula is
T=4B+10C
Sue is placing cookies in bags. She has 10 peanut butter cookies and 16 oatmeal cookies. She will place the same number of cookies in each bag and there will be only one type of cookie in each bag. If sue places the greatest number of cookies possible in each bag, how many bags will she need?
How does sec(x)-1/tan(x) + tan(x)/sec(x)+1 = 2sin(x)/1+cos(x)
Answer in the attachment.
Used:
[tex]\sec x=\dfrac{1}{\cos x}\\\\\tan x=\dfrac{\sin x}{\cos x}\\\\\sin^2x+\cos^2x=1\to\sin^2x=1-\cos^2x[/tex]
How to use a generic rectangle by multiplying 54×32
In a generic rectangle, split 54 into 50 and 4, and 32 into 30 and 2. Set up these numbers around a rectangle divided into four boxes. Each box is the product of the row and column which intersect at the box. Adding up the values in the boxes gives you the product of 54 and 32.
Explanation:To use a generic rectangle in multiplying 54 by 32, you first need to break both numbers down. This is part of a strategy known as partial products.
For example, think of 54 as (50 + 4) and 32 as (30 + 2). Then, set up these numbers around a rectangle, splitting it into four sections or boxes. On one side of the rectangle, put 50 and 4, and on the other side put 30 and 2. Each box represents the product of the row and column that intersect at that box.
Now, calculate the content of each box. For instance, the top left box would have 50 x 30 = 1500. The top right box would have 50 x 2 = 100. The bottom left box would have 4 x 30 = 120, and the bottom right box would have 4 x 2 = 8. The sum of the values inside the boxes (1500 + 100 + 120 + 8) results in 1728, which is the product of 54 and 32.
Using the generic rectangle to multiply numbers is a visual strategy that helps with breaking down more complex multiplication, especially when it involves larger numbers.
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Pls help! will give brainliest(geometry)
2. Linear Pair Postulate
4. 60° + m∠ACB = 180°
5. m∠ACB = 120°
6. definition of obtuse angle
Find the slope between the pair of points using the slope formula. (4,−5) and (0,−13)
The formula of a slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
We have (4, -5) and (0, -13).
Substitute:
[tex]m=\dfrac{-13-(-5)}{0-4}=\dfrac{-13+5}{-4}=\dfrac{-8}{-4}=2[/tex]
Rectangle ABCD is similar to Rectangle WXYZ . The area of ABCD is 30 square inches. Explain how to find the area, x , of WXYZ
The completed statement with regards to the area of the rectangle WXYZ can be presented as follows;
AD/WZ = 1/2, AB/WX = 1/2 Because the ratio of the corresponding side lengths is 1/2, the ratio of the areas is equal to ( 1 / 2 ) to the second power. To find the area, solve the proportion 30/x = 1 / 4 to get x = 120
The steps by which the area of the rectangle WXYZ is found can be presented as follows;
The area of the rectangle ABCD = 30 square inches
AD/WZ = 1/2, and AB/WX = 1/2
The ratio of the corresponding sides in the rectangles ABCD and WXYZ is 1/2
Therefore, scale factor of the lengths of the sides of rectangle ABCD to the lengths of the sides of the rectangle WXYZ is 1/2
The scale factor of the lengths = 1/2
The scale factor of area = (Scale factor for length)²
The scale factor for the area of the rectangles is therefore; (1/2)² = 1/4
Area of rectangle ABCD/(Area of rectangle WXYZ) = 1/4
30/x = 1/4
x is; 30/(1/4) = 120
Area of the rectangle ABCD is about 120 square inches
The complete question found from a similar question on the website can be presented as follows;
Rectangle ABCD is similar to rectangle WXYZ. The area of ABCD is 30 square inches. Explain how to find the area x of WXYZ. AD/WZ = 1/2
AB/WX = 1/2. Because the ratio of corresponding side lengths is 1/2, the ratio of the area is equal to (___/___) to the second power. To find the area, solve the proportion 30/x = __/___ to get x = ___
PLEASE HELP ASAP!!! CORRECT ANSWER ONLY PLEASE!!
Use the imaginary number i to rewrite the expression as a complex number:
The answer to the question is 10 + 9i, B.
When all four transformation types are applied to the same function, which one should be applied first?
The first transformation that is applied first is vertical shift.
What is transformation?A transformation is a general term for four specific ways to manipulate the shape and/or position of a point, a line, or geometric figure.
To apply the transformation on the function the order should be follow is
e than one transformation. Apply the transformations in this order:
1. Start with parentheses (look for possible horizontal shift)
(This could be a vertical shift if the power of x is not 1.)
2. Deal with multiplication (stretch or compression)
3. Deal with negation (reflection)
Hence, the transformation applied is vertical shift.
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In a study about the relationship between marital status and job level, 8,235 males at a large manufacturing firm reported their job grade (from 1 to 4, with 4 being the highest) and their marital status. This two-way table shows the results:
Job Grade Single Married Divorced Widowed Total
1 58 874 15 8 955
2 222 3,927 10 20 4,239
3 50 2,396 34 10 2,490
4 7 533 7 4 551
Total 337 7,730 126 42 8,235
What's the conditional relative frequency among widowed employees of those widowed working at grade 2 or below?
A. 28/42 = 66.67%
B. 34/42 = 80.95%
C. 20/42 = 47.62%
D. 42/4239 = 0.1%
Answer:A)The conditional relative frequency among widowed employees of those widowed working at grade 2 or below= 28/42 = 66.67%
Step-by-step explanation:
In a two way frequency table, conditional relative frequency is it’s a fraction that tells us how many elements of of a group have a certain characteristic.
Here In a study about the relationship between marital status and job level, 8,235 males at a large manufacturing firm reported their job grade (from 1 to 4, with 4 being the highest) and their marital status.
from the given two way frequency table,
Number of widowed employees at grade 1 =8
Number of widowed employees at grade 2=20
∴ Number of widowed employees at grade 2 or below=8+20=28
And the total widowed employees working =42
Now the conditional relative frequency among widowed employees of those widowed working at grade 2 or below=[tex]\frac{\text{Number of widowed employees at grade 2 or below}}{\text{Total widowed employees}}=\frac{28}{42}=0.6667=66.67%[/tex]
Therefore, the conditional relative frequency among widowed employees of those widowed working at grade 2 or below= 28/42 = 66.67%
To calculate the conditional relative frequency among widowed employees for those at grade 2 or below, the number of widowed employees at these grades (28) is divided by the total number of widowed employees (42), yielding a frequency of approximately 66.67%. The answer is option A.
The question asks us to find the conditional relative frequency among widowed employees of those working at grade 2 or below at a large manufacturing firm. To find this, we look at the two-way table provided and sum the number of widowed employees in job grades 1 and 2, which gives us
8 + 20 = 28 widowed employees. We then divide this number by the total number of widowed employees to get the conditional relative frequency:
28/42*100 = 66.67%.
Therefore, among widowed employees, approximately 66.67% work at job grade 2 or below. The answer is option A.
if(x)=x/2-3 and g(x)=4x2+x-4 find (f+g)(x)
[tex](f+g)(x)=4x^2+\frac{3}{2}x-7[/tex]
Answer:
[tex]h(x)=4x^2+\dfrac{3}{2}x-7[/tex]
Step-by-step explanation:
Given:
[tex]f(x)=\dfrac{x}{2}-3[/tex]
[tex]g(x)=4x^2+x-4[/tex]
To find: (f+g)(x)
It is a composite function. Sum of f and g function.
[tex]h(x)=f(x)+g(x)[/tex]
[tex]h(x)=\dfrac{x}{2}-3+4x^2+x-4[/tex]
Combine the like term
[tex]h(x)=4x^2+\dfrac{3}{2}x-7[/tex]
Hence, The sum of f(x) and g(x) would be [tex]h(x)=4x^2+\dfrac{3}{2}x-7[/tex]
Question 6
Solve the following system of equations by using the elimination method.
x + y = 4
2x + 3y = 9
(3, 1)
(-2, 1)
(3, -1)
Answer is (3,1)
x + y = 4
2x + 3y = 9
We multiply the first equation by -2 to eliminate x
-2x - 2y = -8
2x + 3y = 9
-------------------(add both equations)
0 + y = 1
So y =1
Now plug in 1 for y in first equation
x + y = 4
x + 1 = 4
Subtract 1 on both sides
x = 3
Answer is (3,1)
The total daily cost (in dollars) of manufacturing scented candles is given by the function C(b) = 250 + 3b, where b is the number of candles manufactured. Which function represents the average manufacturing cost, A, in terms of b?
The average total cost of manufacturing scented candles is given by the formula A(b) = C(b)/b = (250 + 3b)/b, where b is the number of candles manufactured.
Explanation:The concept of average total cost is key in the calculation. Given the function for total cost of the candles, C(b) = 250 + 3b, the average cost, A, can be calculated by dividing the total cost by the number of candles, b. Therefore, the function representing the average cost, A, in terms of b is given by A(b) = C(b)/b = (250 + 3b)/b. It's important to note that this function also signifies the 'economies of scale', stating as output number increases, the average cost falls.
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How to turn 72 divided by 24 into a fraction
Every time you divide something by something else, you already have a fraction. A fraction is simply a different way to visualize a division. So, the following writings express the same concept:
[tex] 72\div 24 = \dfrac{72}{24} [/tex]
There's no conceptual difference between the two expressions, so you're not actuall "turning" 72 divided by 24 "into" a fraction, you're just writing it in another way.
Triangle XYZ is isosceles with <Z as the vertex angle.
If <X = 5x+33, <Y = 33x+42,
Find <Z
Answer: ∠Z = 117.2°
Step-by-step explanation:
∠X and ∠Y are isosceles so they are congruent (equal to each other)
5x + 33 = 33x + 42
-5x -5x
33 = 28x + 42
-42 -42
-9 = 28x
[tex]\frac{-9}{28} = \frac{28x}{28}[/tex]
[tex]-\frac{9}{28} = x[/tex]
************************************************************
∠X + ∠Y + ∠Z = 180° Triangle angle sum theorem
5x + 33 + 33x + 42 + ∠Z = 180°
38x + 75 + ∠Z = 180°
-75 -75
38x + ∠Z = 105°
38[tex](-\frac{9}{28})[/tex] + ∠Z = 105°
-12.2 + ∠Z = 105°
+12.2 +12.2
∠Z = 117.2°
The local service center advertises that it charges a flat
fee of $50 plus $8 per mile to tow a vehicle. The function
Cx x ( ) = + 8 50 represents the cost C (in dollars) of
towing a vehicle, where x is the number of miles the
vehicle is towed. Identify the slope and y-intercep
The slope of the linear equation C(x) = 8x + 50 is 8, indicating an $8 charge per mile, and the y-intercept is 50, the flat towing fee. Similar service-based costs from other examples also follow this linear model, with their unique slopes and y-intercepts corresponding to the per-hour or per-mile charges and flat fees.
Explanation:The function C(x) = 8x + 50 represents the cost of towing a vehicle, where C is the cost in dollars, and x is the number of miles the vehicle is towed. The slope of this line is 8, which indicates that the cost increases by $8 for every mile the vehicle is towed. The y-intercept is 50, representing the flat fee charged regardless of the distance towed, which is the cost when x (the number of miles) is 0.
Relating to the other examples provided, we can see a pattern in how these equations are structured for service-based costs. For instance, in Aaron's Word Processing Service, the total cost equation is y = 32x + 31.50, where the slope is 32 representing the hourly charge, and the y-intercept is 31.50, the fixed cost. Similarly, for Ethan's household appliance repairs, the equation is y = 20x + 25, with a slope of 20 and a y-intercept of 25. All these models illustrate how fixed fees and variable rates per service hour or mile can be represented in a linear equation format.
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A frost has caused a temporary spike in the price of orange juice. The price is now $10 a gallon. Before the frost, the price was $4 a gallon. How much more expensive is orange juice now? A) 0.25 times b) 1.5 times c) 2.5 times d) 5 times e) 6 times
Write y= -4x² -16x -14 in vertex form.
A) y= -4(x-16)² -14
B) y= 4(x+4)² +2
C) y= -4(x+2)² +2
D) y= (x+2)² -14
B and D are both wrong. B has +4 outside the brackets. It is - 4
D is wrong because there is no 4 of any kind outside the brackets.
A is incorrectly represented inside the brackets as 16. That's not right
That only leaves C. <<<< Answer
Graphs
Notice that the red parabola and the green one are the same thing.
Red: y = -4x^2 - 16x - 14
Green: y = -4(x + 2)^2 + 2
BRAINLIEST!!PLEASE HELP
Given the formula for the perimeter of a rectangle where l represents the length and w represents the width.
2(l + w)
What does the 2 represent in this formula?
A) The 2 represents the perimeter.
B) The 2 represents the two width pairs.
C) The 2 represents the two length pairs.
D) The 2 represents the two sets of length and width pairs.
D the 2 represents the two sets of length and width pairs
2(l + w) = 2l + 2w , that is 2 times length + 2 times width
There are 3 blue marbles, 8 red marbles, 4 green marbles and 5 yellow marbles. What number represents 50% of the marbles?
Theo's mom bought an 8 pack of juice boxes at the store for $4. Find the unit rate for the juice boxes
The measure of an angle is fifty-nine times the measure of a supplementary angle.What is the measure of each angle?
3° and 177°
supplementary angles sum to 180°
let x be an angle then the other angle is 59x
x + 59x = 180
60x = 180 ( divide both sides by 60 )
x = [tex]\frac{180}{60}[/tex] = 3
the angles are 3° and (59 × 3 )= 177°
To find the measures of two supplementary angles where one is fifty-nine times larger than the other, set up an equation x + 59x = 180, and solve for x. The smaller angle is 3 degrees, and the larger angle is 177 degrees.
Explanation:The measure of an angle is fifty-nine times the measure of a supplementary angle. To solve this, we can set up an equation where x is the measure of the smaller angle, and therefore, 59x is the measure of the larger angle. Since angles are supplementary, their sum is 180 degrees. So, our equation is x + 59x = 180. Solving for x, we find that x = 3 degrees and 59x = 177 degrees.
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NEED ANSWER ASAP!!!!!!
3. A local band had a concert and made $1,824 from ticket sales. If each ticket cost $16, how many tickets did the band sell? (Write and solve an equation.)
What are you trying to find?
What is the important information?
Write the equation below. What does the variable represent?
Solve and show work below.
First take not of the total sales and the cost of each ticket.
1,824 and 16
Now setup the equation
16x=1824
To solve this divide 1824 by 16 getting
X=144
x is the total tickets sold
Each ticket (t) costs $16. The band made a total of $1,824 from ticket sales, meaning that 16t=1,824, because the total amount of 16 dollar tickets sold is equal to the total made from ticket sales.
The equation is:
16t=1,824.
You are trying to find the amount of tickets sold by solving the equation.
t stands for the total amount of tickets sold.
Solving the equation:
16t=1,824
Divide both sides by 16.
t=114
114 tickets were sold. 114*16=1,824. This means that when 114 $16 tickets were sold, the band did make $1,824, which shows that 114 is the correct answer.
I hope this helps :)
Five more than twice the square root of four
The phrase 'five more than twice the square root of four' is a mathematical expression which evaluates to 9. The operation uses addition, multiplication, and square root.
Explanation:The student is asking for the result of a mathematical expression combining addition, multiplication, and square root operations. It can be written like "5 + 2(sqrt(4))". The square root of four (sqrt(4)) is two. So, the expression becomes "5 + 2*2", which equals 5 plus 4, resulting in 9. Therefore, five more than twice the square root of four equals nine.
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Which description is correct for the polynomial 4x^2+3x-2?
The first one is correct it is a Quadratic Polynomial.