Answer: [tex]x=-2 \pm \sqrt{29/5}[/tex]
Step-by-step explanation:
[tex]5x^2 + 20x=9\\\\5(x^2 + 4x)=9\\\\5\left((x+2)^2 -4 \right)=9\\\\5(x+2)^2 - 20=9\\\\5(x+2)^2 = 29\\\\(x+2)^2 = \frac{29}{5}\\\\x+2 =\pm \sqrt{29/5}\\\\x=-2 \pm \sqrt{29/5}[/tex]
A textbook store sold a combined total of 460 physics and math textbooks in a week. The number of math textbooks sold was 62 less than the number of physics textbooks sold. How many textbooks of each type were sold?
Number of physics textbooks sold was 261 & number of maths textbooks sold was 199.
Let, the number of physics textbooks sold = x
According to the question,
The number of math textbooks sold was 62 less than the number of physics textbooks sold.
So, Number of maths textbooks sold = x-62
The textbook store sold a combined total of 460 physics and math textbooks in a week.
So, the equation becomes,
(x-62)+x = 460
⇒ x-62+x = 460
⇒ 2x-62 = 460
⇒ 2x = 460+62
⇒ 2x = 522
⇒ x = 522/2
⇒ x = 261
So, 261 physics textbooks were sold
(261-62) = 199 maths textbooks were sold.
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Rafael found an icicle 20 inches long. How long is it in feet? Write your answer as a whole number or a mixed number in simplest form. Include the correct unit in your answer.
Answer:
1 2/3
Step-by-step explanation:
12 inches = 1 foot
So, 20/12 is equal to 1 8/12 foot
remeber to simplify though, so 8/12 simplified is 2/3, so the solution to the problem is 1 2/3 foot
Solve the inequalities by graphing. Select the correct graph. y > 2x y < 3
The solution of the first graph is 1, 2 while the second graph is a straight line.
How to solve for the graphWe have y > 2x
y< x
First you have to replace the inequality by a mathematical operator
y - 2x = 0
y - 3 = 0
equation 1 is of the form
y = mx
The slope here is 2,
to make a graph we need to have x = 1 then the line would pass through
(1,2).
The second equation is simply a second line that would pass through the x axis.
The correct graph is below:
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Which expressions are equivalent to the one below? Check all that apply.
49x
The equivalent expressions are (b) (7 * 7)^x , (c) 7^x * 7^x and (e) 7^(2x)
How to determine the equivalent expressions?The expression is given as:
49^x
Express 49 as 7 * 7
(7 * 7)^x ---- option (b)
Express 7 * 7 as an exponent
7^(2x) ---- option (e)
Express 2x as x + x
7^(x +x)
Expand
7^x * 7^x --- option (c)
Hence, the equivalent expressions are (b), (c) and (e)
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What is the numerator of the simplified sum?
X
x²+3x+2x+1
3
O x+3
O 3x+6
O 4x+6
O 4x+2
Answer:
4x+6
Step-by-step explanation:
x/x^2+3x+2 + 3/x+1
x/(x+1)(x+2) + 3/x+1
make the denominator the same
x/(x+1)(x+2) + 3 (x+2)/(x+1)(x+2)
x/(x+1)(x+2) + 3x+6/(x+1)(x+2)
4x+6/(x+1)(x+2)
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The requried, numerator of the simplified sum is 4x + 6. The option is correct.
What is simplification?Simplification involves applying rules of arithmetic and algebra to remove unnecessary terms, factors, or operations from an expression. The goal is to obtain an expression that is easier to work with, manipulate, or solve. For example, simplifying an algebraic expression might involve combining like terms, factoring, or using the distributive property
The given expression is x/(x²+3x+2) + 3/(x+1). To make the denominators the same, we need to factor the denominator of the first fraction:
x/(x+1)(x+2) + 3/(x+1)
Now we can add the two fractions by finding a common denominator, which is (x+1)(x+2):
x/(x+1)(x+2) + 3(x+2)/(x+1)(x+2)
Simplifying the numerators, we get:
x + 3x + 6 / (x+1)(x+2)
Combining like terms in the numerator, we get:
4x + 6 / (x+1)(x+2)
Therefore, the simplified expression is (4x+6)/(x+1)(x+2).
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Next, create a line through point C that is perpendicular to . Label the intersection between the perpendicular line and as point E. Take a screenshot of the triangle with CE, save it, and insert the image in the space below.
Question : Draw a triangle ABC. Next, create a line through point C that is perpendicular to . Label the intersection between the perpendicular line and as point E. Take a screenshot of the triangle with CE, save it, and insert the image in the space below.
We know that having three sides, three angles, and three vertices, a triangle is a closed, two-dimensional shape. A polygon also includes a triangle.
A line is a straight, one-dimensional figure in geometry that is infinitely long in both directions and has no thickness.
The definition of a perpendicular line in geometry is a pair of lines that meet or intersect at right angles (90°). The Latin word "perpendicularis," which denotes a plumb line, is where the word "perpendicular" first appeared. Two lines, AB and CD, can be written as AB⊥CD if they are perpendicular to one another. The perpendicular nature of the lines is denoted by the symbol.
Here, we have to draw a triangle. Then we have to draw a line CE which intersects AB at point E.
Image is attached below.
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If a translation of T2, –7(x, y) is applied to ΔABC, what are the coordinates of B'?
(1, –5)
(–4, –10)
(–3, –12)
(3, –12)
The coordinates of B' are 3 and -12. so option D is correct (3, -12)
How to find the coordinates of B'?The point P(x, y) may undergo translation to form P' by using a defined translation rule :
If given the rule T(a, b) synch that P undergoes translation on T; then we will have P'(x+a; y+b)
Therefore,
The point B has its coordinates as B(1, - 5)
Translation rule, T(2, - 7) then the coordination of B after undergoing transformation according to T2;
B(1, - 5) - - - > T(2, - 7) - - - - > B'(1+2 ; - 5+-7)
= B'(3, -12)
Hence, the coordinates of B' are 3 and -12. so option D is correct.
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Answer:
(D) (3, –12)
Step-by-step explanation:
If a translation of T2, –7(x, y) is applied to ΔABC, what are the coordinates of B'? The answer is (3, –12).
Which number line represents the solution set for the inequality -1/2x>4?
Answer:
2nd Option
Step-by-step explanation:
Hello!
First, let's solve the inequality for x.
Solve for x[tex]-\frac12x \ge 4[/tex][tex]x \le-8[/tex]Since the symbol is less than or equal to, it will be a closed circle on Point -8 going in the left direction (all values less than -8).
The answer is the Second Number Line.
Find the area of the shaded region.
The outside of the figure is a rectangle that is 24 inches long and 8 inches high. The entire rectangle is shaded except for a triangle with a base of 24 inches and a height of 8 inches.
Answer:
96 sq. inch
Step-by-step explanation:
rectangle has 24 inches × 8 inches dimensions
Hence, Area of Rectangle,
A1=24×8
=192 sq. inch
We, know triangle with base 24 inches and height 8 inches has area exactly half of the rectangle's.
ie. A2=½(24×8)
=96 sq. inch
This was for unshaded region. Then remaining half area is for shaded region.
A1-A2=A
ie. 192-96=96 sq. inch
The area of the shaded region is 96 square inches.
How to determine the area of the shaded regionTo find the area of the shaded region, we need to subtract the area of the triangle from the area of the rectangle.
The area of a rectangle is calculated by multiplying its length by its width. In this case, the rectangle has a length of 24 inches and a width of 8 inches, so its area is 24 inches * 8 inches = 192 square inches.
The area of a triangle is calculated by multiplying its base by its height and dividing the result by 2. In this case, the triangle has a base of 24 inches and a height of 8 inches, so its area is (24 inches * 8 inches) / 2 = 96 square inches.
Now, to find the area of the shaded region, we subtract the area of the triangle from the area of the rectangle:
Shaded Area = Area of Rectangle - Area of Triangle
= 192 square inches - 96 square inches
= 96 square inches
Therefore, the area of the shaded region is 96 square inches.
..
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Aimee is shipping a care package to her friend in college, and she has a piece of
cardboard with which to create a box. A rectangular cardboard sheet measuring 20
inches by 40 inches is to be used by cutting out square corners of measurex inches.
40 inches
XI
X
L.
20 inches
I
Aimee is shipping a care package to her friend in college, the max volume of the box is mathematically given as
V=1539.6007
What is the max volume of the box?Generally, the equation for the height is mathematically given as
V=(20-2x)(40-2x)x
Where x is the height
V=4x^3-120x^2+800x
differentiate v and solve
dv/dx=3x^2-60+200
3x^2-60+200=0
Therefore
x=[tex]\frac{60\pm \sqrt{60^2-4x3x200}}{2*3}[/tex]
x=[tex]\frac{60-34.64}{6}[/tex]
x=4.2265
In conclusion, the second differentiation of v and solve we have
d^2v/dx^2=24-240
Hence
d^2v/dx^2< 0
V=(20-2x(40-2x)x
subsitute x
V=(20-2(4.2265))(40-2(4.2265))(4.2265)
V=1539.6007
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The equation k - 4a = -1 what would k be equal 2
Answer:
K=4a-1
Step-by-step explanation:
K = 4a- 1 (shifting -4a to right side from left)
look at the picture
The end behaviour of the function is When x ---> ∞ , f(x) ---> -∞ , when x ---> -∞ f(x) ----> ∞ , Option D is the correct answer.
What is a Function ?A function is a mathematical statement that related the independent and the dependent variable.
The function given is
f(x) = [tex]\rm -2 \sqrt[3]{x}[/tex]
To understand the end behaviour the graph is plotted
When x ---> ∞ , f(x) ---> -∞
when x ---> -∞ f(x) ----> ∞
Therefore Option D is the correct answer.
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Four less than the product of three and a number
four less than three times that number is:3x-4
is:=
three more than two times the number: 2x+3
Problem to solve:
3x-4=2x+3
3x-2x-4=2x-2x+3
x-4=3
x-4+4=3+4
x=7
Calculated:
3(7)-4=2(7)+3
21-4=14+3
17=17
find x and y please help !!
Answer:
x = 125°y = 75°
Solve for x
One exterior angle (x) equals to sum of two remote interior angles 50°, 75°
x = 50° + 75
x = 125°
Solve for y
Enclosed by parallel lines, the angles equals to 180°
55° + 50° + y = 180°
105° + y = 180°
y = 180° - 105°
y = 75°
Complete the following, using ordinary interest. (Use Days in a year table.) (Do not round intermediate calculations. Round the "Interest" and "Maturity value" to the nearest cent.)
principal 1,000 interest rate 8% Date borrowed March 8 Date repaid June 9
Exact time? interest? Maturity value?
The exact time is 93 days
The interest is $20.38
The maturity value is $1020.38
What is the interest and the maturity value?The interest is a function of the time, amount borrowed and the interest rate.
Interest = amount borrowed x time x interest rate
Time = June 9 - March 8 = 93 days
1000 x 0.08 x (93/365) = $20.38
Maturity value = 1000 + 20.38 = $1020.38
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If we run the following line:
y = int(3 * '4')
what is the value of y?
The value of y in the code segment is 444
How to determine the value of y?The code segment is given as:
y = int(3 * '4')
Evaluate the expression in the bracket
y =int('444')
Remove the bracket
y = 444
Hence, the value of y in the code segment is 444
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Graph the image using the given transformation
The triangle KLM is rotated about origin by 90 degrees to form the triangle K'L'M'.
What is a transformation of geometry?A spatial transformation is each mapping of feature shapes to itself, and it maintains some spatial correlation between figures.
Rotation does not change the size and shape of the geometry.
The triangle KLM is rotated about origin by 90 degrees to create the triangle K'L'M'.
The diagram is given below.
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Which one of the following situations could be represented by a linear function?
A. Number of bacteria is doubled every day.
B. Jacob drives a car and covers the same distance for every hour.
Which value is equivalent to 48 ÷ ((2 + 6) × 2) − 1? Group of answer choices 2 -12 3 3 1/5
The solution to the given expression is 2
Solving equationGiven the equation 48 ÷ ((2 + 6) × 2) − 1
Using PEMDAS
48 ÷ ((8) × 2) − 1
48÷(16) - 1
3 - 1
2
Hence the solution to the given expression is 2
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Which of the following options is an even function?
Using it's concept, an even function is given by:
B. [tex]F(x) = 2\cos{\frac{1}{2}x}[/tex]
What are even functions?Even functions are functions for which the following statement is true for all values of x.
f(x) = f(-x).
The cosine function, without a phase shift, is an example of an even function, hence option B is correct.
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Urgent!!!
If the sum of 5 consecutive positive integers is w , what is the sum of the next 5 consecutive positive integers in terms of w?
A. w+25 B. w+23 C. w+27 D. w+29
The sum of the next 5 consecutive positive integers is w + 25
How to determine the sum of the next numbers?Let the first five numbers be A - E
So, we have:
A + B + C + D + E = W
The next five numbers would be
A + 5, B + 5...... E + 5
So, we have:
Sum = A + 5 + B + 5 + C + 5 + D + 5 + E + 5
Evaluate the like terms
Sum = A + B + C + D + E + 25
Substitute A + B + C + D + E = W
Sum = W + 25
Hence, the sum of the next 5 consecutive positive integers is w + 25
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A manufacturer of bolts has a quality-control policy that requires it to destroy any bolts that are more than 2 standard deviations from the mean. The quality-control engineer knows that the bolts coming off the assembly line have mean length of 7 cm with a standard deviation of 0.10 cm. For what lengths will a bolt be destroyed?
The bolt of length more than 7.2 cm will be destroyed.
Mean refers to the central tendency of a data set. It is calculated by taking average of the data.
It is given that
any bolts that are more than 2 standard deviations from the mean will be destroyed
Mean = 7 cm
Standard Deviation is = 0.10 cm
The length of the bolt which is not acceptable is = Mean + 2 Standard Deviation.
Length = 7 + 2 * 0.10
Length = 7.2 cm
Therefore if the bolt of length more than 7.2 cm will be destroyed.
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A count of the number of even numbers obtained from a random number
generator would best be modelled by what distribution?
A. geometric
B. Poisson
C. binomial
D. normal
Considering that there are only two possible outcomes for each number, and a fixed number of numbers, the distribution is:
C. binomial.
What distribution models the situation?We have that:
For each number on the sequence, there are only two possible outcomes, either it is even or it is odd.A number being even is independent of any other number.The sequence has a constant cardinality.Hence the binomial distribution should be used and option B is correct.
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What is the greatest common factor of 8 and 24?
Answer:
The G.C.F of (8,24) is 8
Step-by-step explanation:
look at the attachment above ☝️
Find the negation of the proposition
In accordance with propositional logic, quantifier theory and definitions of simple and composite propositions, the negation of a implication has the following equivalence:
[tex]\neg (\exists \,x\, (P(x) \implies Q(x))) \iff \forall \,x (\neg \,Q(x) \,\land \,P(x))[/tex] (Correct choice: iii)
How to find the equivalent form of a proposition
Herein we have a composite proposition, that is, the union of monary and binary operators and simple propositions. According to propositional logic and quantifier theory, the negation of an implication is equivalent to:
[tex]\neg (\exists \,x\, (P(x) \implies Q(x))) \iff \forall \,x (\neg \,Q(x) \,\land \,P(x))[/tex]
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What is the value of the expression shown 7 + (2 + 6)^2 divided by 4 x (1/2)^4
The value of the expression is 188
How to determine the value
Given the expression
= [tex]\frac{7 + (2 + 6)^2}{4 * \frac{1}{2}^4}[/tex]
Expand the bracket
= [tex]\frac{7 + 4 + 36}{ 4 * 1/16}[/tex]
We have
= [tex]\frac{47}{1/4}[/tex]
The denominator becomes the multiplier, we have
= 47 × 4
= 188
Thus, the value of the expression is 188
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You need a 25% alcohol solution. On hand, you have a 225 mL of a 35% alcohol mixture. How much pure water will you need to add to obtain the desired solution?
You will need____ mL of pure water to obtain___ mL of the desired 25% solution.
You will need 252 mL of pure water to obtain 0.25 mL of the desired 25% solution.
If we add X amount of water to the solution, the total amount of solution is 375+X.
And that must be 25%.
What is the amount of solute in the current solution?
The amount of solute in the current amount of solution is
0.6(375)=225.
That amount won't change by adding water.
So we set up the equation so that the amount of solute divided by the new total amount of solution equals 25%.
225/(375+X)=0.25
cross multiply and solve
225/0.25=(375+X)
900-375=X
X=525
The pure water will you need to add to obtain the desired solution is 525.
Then we check 225/(375+525)=0.25.
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the number of users on a new social media website can be modeled by the function f(t) = 12(1.2)^t , where t is the time, in weeks, since the site was launched. Find and interpret the average rate of change on the interval (3,6) round to the nearest whole number
The number of users on new social media site increase as 21, 25, 30, 36,
The average rate of change is given by 5.
The function f (t) = 12 (1.2) ^t show how the rate of users increases in weeks.
Average of the values is the ratio of total sum of values to the number of values.
Here, f (t) = 12 (1.2) ^t is defined in the range of (3, 6)
So the value of the function in its range is given by
1) f (3) = 12 (1.2) ^3
= 21
2) f (4) = 12 (1.2) ^4
= 25
3) f (5) = 12 (1.2) ^5
= 30
4) f (6) = 12 (1.2) ^6
= 36
Now the average rate of change = sum of all difference in values at every week / max. Difference in interval of range(3,6)
= f(6)-f(3)/6-3
= 15/3
= 5
Thus required average change of rate is 5.
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What is the image of A(5,6) under R90°?
For the coordinate (5,6), the 90 degree rotation will be equivalent to (-6, 5)
Rotation of imagesRotation is a technique used to change the position of an object on an xy-plane.
If the coordinate point (x, y) is rotated 90degrees, the rule will be:
(x, y) -> (-y, x)
For the coordinate (5,6), the 90 degree rotation will be equivalent to (-6, 5)
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I forgot how to solve this equation for both answers already :( I will be very much appreciated if I could get a step by step clear solution for both answers!
The equations for the lines tangent to and normal to the curve at the point (2, 4) are y = 4 · x - 4 and y = (-1/4) · x + 9/2, respectively.
How to derive the equation of lines tangent and normal to a point of a curve
In this question we must use the definitions of linear function, tangent and normal lines and differential calculus to determine the equations needed. The slope of the tangent line (m) is found by implicit differentiation:
2 · x + 2 · y · y' - 3 · (y + x · y') = 0
2 · x + (2 · y - 3 · x) · y' - 3 · y = 0
(2 · y - 3 · x) · y' = 3 · y - 2 · x
y' = (3 · y - 2 · x)/(2 · y - 3 · x)
By (x, y) = (2, 4)
m = (3 · 4 - 2 · 2)/(2 · 4 - 3 · 2)
m = 4
And the slope of the normal line is:
m' = -1/m = -1/4
Lastly, we obtain the intercept for each line:
Tangent line
b = 4 - 4 · 2
b = 4 - 8
b = - 4
The equation of the line tangent to the curve at the point (2, 4) is y = 4 · x - 4.
Normal line
b = 4 - (-1/4) · 2
b = 4 + 1/2
b = 9/2
The equation of the line normal to the curve at the point (2, 4) is y = (-1/4) · x + 9/2.
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