Answer: (a) The equation 2x^2 + 4x + 3 = k has real roots if and only if its discriminant, which is 4^2 - 4(2)(3), is nonnegative. The discriminant is equal to 0, so the equation has real roots if and only if k = 3.
(b) For the equation f(x) = kx^2 + kx + 1 to have real roots, its discriminant, which is 1 - 4k, must be nonnegative. This gives us k >= 1/4. The range of values of k for which the equation has real roots is k >= 1/4.
(c) When k = -2, the equation f(x) = -2x^2 - 2x + 1 = 0. Solving this equation using the quadratic formula, we get x = (-1 ± √(1 + 8))/4. So, the roots are x = (-1 ± √(9))/4 = (-1 ± 3)/4 = 1/2, -2.
focus (4,-3) directrix y=-3 what’s the equation for the parabola
The equation of the parabola is x² - 8x + 16 = 0
What is the equation of the parabolaTo find the equation of the parabola with focus (4, -3) and directrix y = -3, we can use the definition of a parabola.
A parabola is the set of all points that are equidistant to the focus and the directrix.
The distance from a point (x, y) to the directrix y = -3 is simply the vertical distance between the point and the line, which is |y - (-3)| = |y + 3|.
The distance from a point (x, y) to the focus (4, -3) can be found using the distance formula:
d = √[(x - 4)² + (y + 3)²]
Since the point is equidistant to the focus and the directrix, we have:
d = |y + 3|
Squaring both sides, we get:
(x - 4)² + (y + 3)² = (y + 3)²
Simplifying, we get:
(x - 4)² = 0
Thus, the equation of the parabola is simply:
x² - 4x - 4x + 16 = 0
x² - 8x + 16 = 0
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Show how 15 + 5n and 5(3 + ) are equivalent
Answer: 5n+30
( The problem was incomplete on my end )
( Hope this helps )
In Youngtown, the population reached 4420 people in 1993. In 1998, there were 5600 residents. What was the population in the year 2000?
6062
6068
6058
6072
NEED HELP ASAP
Answer:
6,072
Step-by-step explanation:
5,600 - 4,420 = 1,180
They gained 1,180 people over 5 years. Now we need to find how many people they gain every year (roughly).
Since it's 5 years, we are going to divide.
1,180 / 5 = 236
They gained 236 people over 1 year. Since 1998 is two years is before 2000, we're going to multiply 236 by 2.
236 x 2 = 472
Now we add this to the 1998 population.
5,600 + 472 - 6,072
The population in the year 2,000 is 6,072
Solve the following
i} p+ 3 = 205 ii} 2x/7 = 4 iii} 3y - 6 = 9
Answer:
[tex]1) p = 205 - 3 = 202 \ 2)2x \div 7 = 4 \\ 2x = 28 \\ x = 14 \\ \\ 3)3y - 6 = 9 \\ 3y = 9 + 6 \\ 3y = 15 \\ y = 5[/tex]
Express the product shown as a fraction in simplest form: -2/5 . 2/5
The simplified product of - (2/5).(2/5) is equal to - 4/25.
What is a fraction?A fraction is written in the form of p/q, where q ≠ 0.
Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.
A product of two fractions - 2/5 and 2/5 is given.
We know in the product of fractions numerator is multiplied by the numerator and the denominator is multiplied by the denominator.
Therefore, - (2/5).(2/5).
= - (2×2)/(5×5).
= - 4/25.
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What is the arc measure of major arc BAC in degrees? no
Answer: 360 = 7n + 12 + 13x — 16 + 6xx = 146 * 14 = 84°
HELLLPPP ILL MARK BRAINLIEST
Answer:
Below
Step-by-step explanation:
2/3 x = 9 - 2(-1/3 x +3 ) Expand the parentheses
2/3 x = 9 - -2/3x -6 simplify the R side a bit
2/3 x = 9 + 2/3x - 6
2/3 x = 3 + 2/3 x subtract 2/3 x from both sides
0 = 3 <===== obvously not true, NO SOLUTION
a) complete the table of values for y = 2x^2 + x
b) Which of the three curves drean matches the graph of y = 2x^2 + x
Answer:
a)
x y
-2 6
-1 1
0 0
1 3
2 10
b) Curve B matches all the values.
Step-by-step explanation:
See attached worksheet.
a) Enter the x values into the equation to find the corresponding values of y.
b) Mark these x,y values on the graph to identify the curve that matches all points. That is curve B.
Gerry is shopping for clothes at the mall. He buys 4 new shirts for $6. 0. What is the unit rate for the shirts
The unit rate of the shirt is $1.50 per shirt.
Unit rate is a measure of the relationship between two quantities, where the second quantity is expressed as a unit of one. It is often expressed as a ratio, with the unit of the second quantity being 1. Unit rate helps us compare different quantities and make sense of their relative sizes.
The unit rate for the shirts can be calculated by dividing the total cost by the number of shirts. In this case, the unit rate would be:
$6.00 ÷ 4 shirts = $1.50/shirt
So, the unit rate for the shirts is $1.50 per shirt.
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Create a flowchart proof of a two column proof. If HG = KG and HI || KJ, prove that HI = KJ
Answer:
Both triangles are congruent by S A S congruent.
Step-by-step explanation:
Congruent triangles:In ΔHIG & ΔKJG,
HG = KG Given
∠GHI ≅ ∠GKJ Alternate interior angles, HI // KJ and KH is transversal
HI = KJ Given
ΔHIG ≅ ΔKJG Side Angle Side congruency
Use the following information to work out the fraction.
In its simplest form, it is 1/1.
The numerator is even, but the denominator is odd.
The denominator is more than 40 and less than 60.
The fraction cannot be worked out, given the information on the numerator being even and the denominator being odd.
How to work out the fraction ?When fractions are said to be in their simplest forms as 1 / 1, then it means that both the numerator and the denominator are the same. This way, they can divide each other and leave one.
For such a situation to happen, the numbers will have to either both be even numbers, or odd numbers. The numerator cannot be even while the denominator is odd.
This fraction therefore cannot be worked out.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
Fred and his workout partner are lifting weights together, doing many sets of each exercise.
On a certain exercise, Fred is using a 19-kilogram bar, increasing the amount of weight he lifts by 4 kilograms on each set. His partner, meanwhile, started out using a 15-kilogram bar and is upping the weight by adding 6 kilograms on every set. Eventually, Fred and his workout partner will be lifting the same amount, and will take turns using the same barbell.
How much weight will they be lifting then? How many sets will they have completed?
The system of equations is:
y = 19 + 4*x
y = 15 + 6x
And the solution is x = 2 and y = 27.
How to write and solve the system of equations?
We know that red is using a 19-kilogram bar, increasing the amount of weight he lifts by 4 kilograms on each set. Then after x sets he will be lifting:
y = 19 + 4*x
kilograms.
And his partner, meanwhile, started out using a 15-kilogram bar and is upping the weight by adding 6 kilograms on every set. Here we can write:
y = 15 + 6x
Then we need to solve the system of equations at:
y = 19 + 4*x
y = 15 + 6x
Using substitution, wecan write:
19 + 4x = 15 + 6x
19 - 15 = 6x - 4x
4 = 2x
4/2 = x
2 = x
They will lift the same weight after 2 sets, and the weight is:
y = 19 + 4*2
y = 19 + 8 = 27
They will be lifting 27 kilograms.
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find the radius and diameter of a circle whose area is 15400 mm²
Solution:
Area=15400mm^2
Now,
Area=pi×r^2
or,15400=22/7×r^2
or,15400×7=22×r^2
or,107800=22r^2
or,107800/22=r^2
or,4900=r^2
or,r=70
Also,
Diameter (d)=r×2
=70×2
=140
Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result.
Vertex: (3, 1); Point: (5, −7)
The required standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point is y = -2x² + 12x - 17.
What is the equation?The equation is the relationship between variables and represented as y = ax + b is an example of a polynomial equation.
To find the quadratic function that has the vertex (3, 1) and passes through the point (5, -7), we can use the vertex form of the quadratic function: y = a(x - h)² + k, where (h, k) is the vertex.
First, we substitute the vertex (5, -7) into the vertex form to find a:
y = a(x - 3)² + 1
-7 = a[2]² + 1
a = -2
Since a = -2, the standard form of the quadratic function is:
y = -2x² + bx + c
Substitute the point (5, -7) into the standard form y = -2x² + bx + c to find b and c:
-7 = -2x² + b(5) + c
-7 = -50 + 5b + c
43 = 5b + c - - - - -(1)
Substitute the point (3, 1) into the standard form y = -2x² + bx + c to find b and c:
1 = -18 + b(3) + c
19 =3b + c - - - - -(2)
Solving equations 1 and 2 we get
43 - 19 = 2b
b = 12 and c = -17
Put a,b and c in the standard equation,
y = ax² + bx + c
y = -2x² + 12x - 17
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−3/4(x+2)=[tex]-3/4(x+2)=6[/tex]6
The value of x for the equation is -10.
What is simplification?To simplify simply means to make anything easier. In mathematics, simplifying an equation, fraction, or problem means taking it and making it simpler. Calculations and problem-solving techniques simplify the issue. By eliminating all common factors from the numerator and denominator and putting the fraction in its simplest/lowest form, we can simplify fractions.
Given equation,
-3/4(x + 2) = 6
both sides multiply by -4/3
-3/4(x + 2) * -4/3 = 6 * -4/3
x + 2 = -8
subtract 2 from both sides
x + 2 - 2 = -8 - 2
x = -10
Hence the value of x is -10.
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4. VOLUME The volume of one column of the Lincoln Memorial is approximated by the expression (x³ +37x²-110x + 400). If the height of the column is x + 40 feet, find the expression that approximates the area of the base of the column in terms of x and pi.
To find the area of the base of the column, we need to find the expression for the radius of the column, r, in terms of x.
The volume of a column can be calculated as V = πr²h, where r is the radius, h is the height, and π is pi.
Using the given expression for the volume of the column, we can write:
πr²h = x³ + 37x² - 110x + 400
Since h = x + 40, we can substitute to get:
πr²(x + 40) = x³ + 37x² - 110x + 400
Dividing both sides by π(x + 40) gives us:
r² = (x³ + 37x² - 110x + 400) / (π(x + 40))
Finally, taking the square root of both sides gives us the expression for the radius in terms of x:
r = √((x³ + 37x² - 110x + 400) / (π(x + 40)))
So, the expression that approximates the area of the base of the column in terms of x and π is:
A = πr² = π √((x³ + 37x² - 110x + 400) / (π(x + 40)))²
Which expression does not have a solution of 12, if w = 4?
Options :
3 w
w + 8
4 w
16 - w
Juan went to the store to buy a plant for his mother, but forgot to ask which kind to get. Out of 30 plants, six of them are a kind that his mother would want. If Juan chooses one randomly, what is the probability that he will choose a kind that his mother would want? You will need to change your probability to a decimal and then to a percent.
The probability that Juan will choose a kind of plant his mother would want is 6/30, which is equal to 0.2 or 20%
To calculate the probability that Juan will choose a kind of plant his mother would want, out of 30 plants, six of them are a kind that his mother would want, first divide the number of plants his mother would want (6) by the total number of plants (30). This will give you a decimal, 0.2. To turn this decimal into a percent, multiply it by 100. This will give you 20%, or a 20% chance of Juan choosing a kind of plant his mother would want. To summarize, therefore, the probability that Juan will choose a kind of plant his mother would want is 0.2 or 20%.
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What is the solution to this equation?
X/4= -12
A. x = -48
OB. x=3
OC. x= -3
OD. x = 48
Answer:
A. x = -48
Step-by-step explanation:
x/4 = -12
x = (-12)(4) = -48
HELP WHAT IS THE VALUE OF TAN A
Answer: 4 / 3
Step-by-step explanation:
Soh Cah Toa
In this case, we'll use Toa
Tangent = Opposite / Adjacent
Tan A = 40 (opposite) / 30 (adjacent)
Tan A = 40 / 30 = 4 / 3 (simplified)
A farmer takes 250 chickens to be sold at a market. In the first hour he sells 8% of his chickens. In the second hour he sells 10% of those that were left. How many chickens has he sold in total?
Answer:
43 chickens
Step-by-step explanation:
1) 250*8%=20
2) 250-20=230
3) 230*10%=23
4) 20+23=43
PRE CALC PEOPLE HELP PLSSSSS
Find three additional polar representations of the point (2, 3π/4)
using −2 < < 2.
(r, ) = smallest value?
(r, ) = ?
(r, ) = largest value?
To find three additional polar representations of the point (2, 3π/4), we can add or subtract 2πk to the angle, where k is an integer. Using this method, we get:
(2, -5π/4) - subtract 2π
(2, 11π/4) - add 2π
(2, -3π/4) - subtract 2π
To find the smallest and largest values of the angle, we need to consider the interval −2π < θ < 2π. The given angle of 3π/4 lies in the second quadrant. Adding or subtracting 2π to this angle will give us angles in the third and fourth quadrants. Therefore, the smallest value of the angle is −3π/4, and the largest value is 5π/4. Thus, the polar representations are:
(2, 3π/4)
(2, -5π/4)
(2, 11π/4)
(2, -3π/4)
(2, 5π/4)
Need help on this question!
There are 4 options:
(-7,1)
(-3,-1)
(-1,-3)
(3,3)
Thanks
Answer:
(-1,-3)
Step-by-step explanation:
See the attached worksheet. First step is to apply the transformation. Modifiy the x and y points as per the instructions. The transformed triangle is shown in blue and the coordinates are listed. As the directions indicate, it is slipped down by 2 and to the left by 3 (y-2 and x-3).
Next, convert the x values so that they are reflected across the y axis. The table shows that all x values are the negative of their original values. The reflected triangle is shown in red.
The new point Z is (-1,-3).
q. The graph shows the change in temperature
in Monterey, CA over several hours. Is the
between the number of hours and
relationship between
temperature linear or non-linear? Explain.
Answer:
Non-linear
Explanation below
Step-by-step explanation:
The relationship is clearly non-linear
If it where a linear relationship then the temperature change per hour would be constant throughout the time period when the observations were made.
The resultant graph would be a straight line
Here the graph rises at a decreasing rate until it almost flattens out. This means the temperature reaches a maximum and stays at that maximum.
Consider the error formulas.
|E| ≤
(b − a)3
12n2
[max |f ''(x)|], a ≤ x ≤ b
Trapezoidal Rule
|E| ≤
(b − a)5
180n4
[max |f (4)(x)|], a ≤ x ≤ b
Simpson's Rule
Use these to estimate the errors in approximating the integral, with n = 4, using the Trapezoidal Rule and Simpson's Rule.
6 1 (x − 1)2 dx 2
Trapezoidal Rule: [tex](b-a)^3 / 96n^2,[/tex] Simpson's Rule: [tex](b-a)^5 / 2880n^4\\[/tex]. Both approximations have error of 0.0208.
For the Trapezoidal Rule, the formula for estimating the error is |E| ≤ (b − a)3/12n2[max |f ''(x)|], a ≤ x ≤ b. In this case, the integral is from x=1 to x=3, so b-a = 2. Plugging in all the values, we get [tex]|E| ≤ 2^3/12(4)^2(2)[/tex] or |E| ≤ 0.0208. For the Simpson's Rule, the formula for estimating the error is |E| ≤ (b − a)5/180n4[max |f (4)(x)|], a ≤ x ≤ b. Again, b-a = 2. Plugging in all the values, we get[tex]|E| ≤ 2^5/180(4)^4(2)[/tex] or |E| ≤ 0.0208. Therefore, the error in approximating the integral with n = 4, using the Trapezoidal Rule and Simpson's Rule, is 0.0208.
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help me baaaaaaaaaaaaaaaalls
Answer:
The correct scale faction the first pentagon dilated by to create the second pentagon is 1.25.
Step-by-step explanation:
Since the second pentagon is a dilation of the first pentagon, we can conclude these pentagons are similar, meaning the shape retained all of its previous angles, and all the corresponding sides are dilated by a common scale factor. Since we know the values of a pair of corresponding sides, we can use those to find the scale factor of dilation.
The corresponding side of the original pentagon is 4 units, and the corresponding side of the dilated pentagon is 5 units. Since the first pentagon dilated to become the second, in order to solve the scale factor the pentagon dilated by, we'd have to divide the side given on the second pentagon by the corresponding side of the first, which will be 5 ÷ 4 = 1.25. Therefore, that will be the scale factor the pentagon dilated by.
Have a great day! Feel free to let me know if you have any more questions :)
Jason purchased a car $22,995. According to his research, this make and model of a car loses all of it's marketable value after 9 years. If this car depreciates in a straight line form, what are the coordinates of the intercepts of the depreciation equation?
The coordinates of the intercepts of the depreciation equation are (9, 0) and (0, 22,995).
What are Intercepts?The x intercept is the x coordinate of a point where the line touches the X axis.
The y intercept is the y coordinate of a point where the line touches the Y axis.
Here, there are two variables.
The independent variable x is the number of years.
The dependent variable y is the value of the car.
x intercept is the year when the value of the car is 0.
Given that, model of a car loses all of it's marketable value after 9 years.
x intercept = 9
y intercept is the value of the car in the 0th year.
Initial value of the car = $22,995
y intercept = 22,995
Hence the intercepts are (9, 0) and (0, 22,995).
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last year, 20 percent of all teenagers purchased a new iphone. this year, a sample of 260 randomly chosen teenagers showed that 39 had purchased a new iphone. to test whether the percentage has decreased, the p-value is:
The p-value is 0.063, which indicates that there is not strong evidence to reject the null hypothesis that the percentage has stayed the same.
The p-value can be calculated by using a one-proportion z-test. The null hypothesis would be that the percentage has stayed the same from last year, and the alternative hypothesis is that the percentage has decreased.
The formula for the z-score is (x-p)/√(p(1-p)/n), where x is the observed proportion (39/260), p is the expected proportion (0.20), and n is the sample size (260). The p-value is then calculated using the z-score and the cumulative distribution function.
Therefore p-value, in this case, is 0.063, which indicates that there is not strong evidence to reject the null hypothesis that the percentage has stayed the same.
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A gardener has 600ft of fencing to fence in a rectangular garden. One side of the garden is boarded by a river and so it does not need any fencing.
What dimensions would guarantee that the garden has the greatest possible area?
Shorter side:
Longer side:
Greatest possible area:___ft^2 (square feet)
Answer:
To find the dimensions that would guarantee the greatest possible area, we can use the formula for the area of a rectangle, which is A = l * w, where l is the length and w is the width.
Since one side of the garden is already boarded by a river, we only need to use 600ft of fencing for the other three sides. This means that the length and width of the garden must add up to 600ft. We can write this as:
l + w = 600
Since we want to find the dimensions that would give us the greatest possible area, we want to maximize l * w. Using the formula for the area of a rectangle, we can write this as:
A = l * w = l * (600 - l) = 600l - l^2
To find the maximum area, we can differentiate this equation and set it equal to zero:
dA/dl = 600 - 2l = 0
Solving for l, we get:
l = 300/2 = 150ft
So, the shorter side is 150ft and the longer side is 450ft (600 - 150).
Finally, we can use the formula for the area of a rectangle to find the greatest possible area:
A = 150 * 450 = 67,500ft^2
Therefore, the greatest possible area for the garden is 67,500ft^2 when the shorter side is 150ft and the longer side is 450ft.
Step-by-step explanation:
PLEASE HELP !!!!!
Function g is graphed here. If function f is the parent exponential function (). what is the equation of transformed function g in terms of f?
Therefore , the solution of the given problem of function comes out to be f(x) = 1.1f(x + 1) is the solution for the graph.
What is function?The study of mathematics includes looking at numbers, their variables, the world around us, buildings, and both real and imagined places. An illustration of the relation between the quantity of inputs and the appropriate outputs for each is called a function. A function, simply described, is a group of inputs who, when combined, provide distinct outputs for each input. Every function has a city, province, or scope—also referred to as a domain—assigned to it. Functions are usually indicated with the letter f. (x). The input contains an x.
Here,
Given:
The parent function f(x) is translated left by one unit and slightly stretched vertically to form g(x).
=>g(x) is:
=> f(x) = 1.1f(x + 1)
Thus , f(x) = 1.1f(x + 1) is the given function which is graphed here.
Therefore , the solution of the given problem of function comes out to be f(x) = 1.1f(x + 1) is the solution for the graph.
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