The solutions to the equation [tex]p^2+4p=1[/tex] using completing the square method are [tex]-2- \sqrt{5}[/tex] and [tex]-2+ \sqrt{5}[/tex]
The given equation is:
[tex]p^2+4p=1[/tex]
Add [tex](\frac{4}{2})^2[/tex] to both sides of the equation
[tex]p^2+4p+(\frac{4}{2})^2= 1 + (\frac{4}{2})^2\\\\p^2+4p+2^2=1+2^2\\\\(p+2)^2=1+4\\\\(p+2)^2=5\\\\[/tex]
Find the square root of both sides
[tex]p+2 = \pm\sqrt{5}\\\\p=-2\pm sqrt{5} \\\\p = -2 - sqrt{5}\\\\p=-2+ \sqrt{5}[/tex]
The roots of the equation are [tex]-2- \sqrt{5}[/tex] and [tex]-2+ \sqrt{5}[/tex]
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The equation p² + 4p = 1 can be solved by first rearranging to p² + 4p - 1 = 0, completing the square, simplifying and taking square roots. The solutions are p = -2 + sqrt(5) and p = -2 - sqrt(5).
Explanation:To solve the quadratic equation p² + 4p = 1 by completing the square, we first arrange the equation with the quadratic and linear terms on one side and the constant on the other:
p²+ 4p - 1 = 0.
Next, we complete the square by adding the square of half the coefficient of p to both sides of the equation. The coefficient of p is 4, so half of it is 2, and squaring it gives 4. So, we have:
p² + 4p + 4 = 1 + 4
This simplifies to:
(p + 2)² = 5
Then, we take square roots of both sides:
p + 2 = ± sqrt(5)
Which further simplifies to:
p = -2 ± sqrt(5)
So, the solutions for this quadratic equation are p = -2 + sqrt(5) and p = -2 - sqrt(5).
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what is 230000 in standard form
Final answer:
To write 230,000 in standard form, place the decimal point after the first non-zero digit. For 230,000, the standard form is 2.3 x 10⁵.
Explanation:
To express the number 230,000 in standard form, you simply write it as it is, moving the decimal point to the right until only one non-zero digit appears before the decimal. In this case, 230,000 in standard form is 2.3 x 10⁵.
BRAINLIEST!
Henry's rocket is designed to launch to a height of 100 feet in the air. The measurements so far have the launch height (h) varying no more than 4 feet. Which inequality below represents this scenario?
|h − 4| less than or equal to 100
|h + 4| greater than or equal to 100
|h − 100| less than or equal to 4
|h + 100| greater than or equal to 4
Answer:
|h − 100| less than or equal to 4
Step-by-step explanation:
We know the difference between h, the actual height, and the designed height, 100, is no more than 4.
This means that the actual height h could be up to 4 more than the designed height or up to 4 less than the designed height. This is why we use an absolute value inequality; this considers just the distance of 4 regardless of direction.
Since it can be "no more than" 4, we use less than or equal to; this gives us
|h − 100| less than or equal to 4
Sheila is building a model for class. It's a tower that's designed to withstand high winds. She can use a maximum of 700 support beams and knows that each floor will require 22 beams. Sheila will track the number of beams that remain as she adds floors.
She can predict the number of beams that remain by using this function: remaining beams (y) is function of floor number (x).
How many beams will remain after she built 6 floors? Y=700-22(x)
Anyone wanna help me please?!
Allen has a photo that is 6 in by 8 in. What will the dimensions of the photo be if he scales it down by a factor of one-half ?
3 in by 4 in
6 in by 8 in
12 in by 16 in
1.5 in by 2 in
Help me please its math under 10 minutes!!!
Deon is saving money to buy a game. so far he has saved $20 , which is four-fifths of the total cost of the game. how much does the game cost?
Evaluate the determination for the following matrix [144] [522] [155]
Which ordered pair (a,b) is a solution to the given system of linear equations.
Answer:
A. (-13,-3)
Step-by-step explanation:
Solve the system of two equations:
[tex]\left\{\begin{array}{l}-2a+3b=14\\a-4b=3\end{array}\right.[/tex]
From the second equation:
[tex]a=3+4b[/tex]
Substitute it into the first equation:
[tex]-2(3+4b)+3b=14\\ \\-6-8b+3b=14\\ \\-8b+3b=14+6\\ \\-5b=20\\ \\5b=-20\\ \\b=-4[/tex]
So,
[tex]a=3+4b=3+4\cdot (-4)=3-16=-13[/tex]
The solution is (-13,-4)
Answer:
A
Step-by-step explanation:
6.
The sum of two consecutive odd integers is 236. What is the smaller integer? (4 points)
a) 115
b) 117
c) 119
d) 113
What is the measure of an exterior angle in a regular dodecagon?
HELP PLZ!!
Solve for x: −3(x + 3) = −3(x + 1) − 5. (1 point)
6
−6
All real numbers
No solution
Marc earned $1,650 per month as an assistant technician at a recording studio. His new job pays $9.80 per hour with time and a half for all hours over 36 hours per week. How many hours of overtime per week will he need to work to earn the same amount per week as his current job?
Marc will need to work just over 4 hours of overtime each week at his new job, which pays $9.80 per hour and $14.70 per hour for overtime, to earn the equivalent of his previous monthly income of $1,650.
To determine the number of overtime hours Marc needs to work at his new job to earn the same monthly income he did at his previous job, we'll first calculate his weekly income at his old job, then use the overtime pay rates at the new job to find the equivalent number of hours needed.
Marc previously earned $1,650 per month. Assuming a standard 4-week month, his weekly income was:
$1,650 per month / 4 weeks per month = $412.50 per week.
At his new job, Marc earns $9.80 per hour with time and a half for all hours over 36 hours a week. This means his overtime pay is 1.5 times his hourly rate, which is:
$9.80 per hour x 1.5 = $14.70 per hour for overtime.
Marc's income for the first 36 hours at his new job would be:
36 hours x $9.80 per hour = $352.80
To earn the equivalent of his old weekly salary, we must make up the difference with overtime pay:
$412.50 (old weekly income) - $352.80 (new regular pay) = $59.70 needed from overtime.
To determine the overtime hours needed to make $59.70: $59.70 / $14.70 per hour = approximately 4.06 hours.
Therefore, Marc will need to work just over 4 hours of overtime each week to earn the same amount he did at his previous job.
Francis has five more dollars than Julia. Together they have at most $45. Let x represent the amount of money Francis has. Which inequality can be used to find the possible amounts of money that Francis has?
Answer:
B) 2x - 5 ≤ 45
Step-by-step explanation:
Can someone help me with this please? And can you explain why that answer is right
The greatest common factor of 4a(2) b and 6ab(3)
The greatest common factor of the expressions 4a^2b and 6ab^3 is 2ab.
The question seeks to determine the greatest common factor of the expressions 4a2b and 6ab3. To find this, one must look at the common factors of the numerical coefficients (4 and 6) and the variables raised to their respective powers (a2 and a, b and b3). The numerical common factor of 4 and 6 is 2. The lowest power of 'a' common to both terms is a1 and for 'b' is b1. On doing this we get 2ab.
Points d, e, and f are plotted on a number line. the coordinate of point f is 5. points d and e are the same distance from point f. what could be the coordinates of d and e?
a. d = –7, e = –1
b. d = –2, e = 12
c. d = 0, e = 5
d. d = 8, e = 11
The coordinates of d and e b. d = –2, e = 12.
To find the coordinates of points D and E, we need to remember that they are the same distance from point F, which has a coordinate of 5. This means D and E are equidistant from F on the number line. Let's check each option to see which one satisfies this condition:
Option a: d = -7, e = -1
The distance from -7 to 5 is 12.
The distance from -1 to 5 is 6.
Since the distances are not equal, this option is incorrect.
Option b: d = -2, e = 12
The distance from -2 to 5 is 7.
The distance from 12 to 5 is also 7.
Since the distances are equal, this option is correct.
Option c: d = 0, e = 5
The distance from 0 to 5 is 5.
The distance from 5 to 5 is 0.
Since the distances are not equal, this option is incorrect.
Option d: d = 8, e = 11
The distance from 8 to 5 is 3.
The distance from 11 to 5 is 6.
Since the distances are not equal, this option is incorrect.
Therefore, the correct answer is option b: d = -2, e = 12.
The solution to the equation 4(x – 7) = 64 is shown. Which step is a result of combining like terms?
A) Step 2
B) Step 3
C) Step 4
D) Step 5
Mrs. Noyes fills up a 5 gallon can of gas from the McIntosh’s local convenience store. They are selling gas at $2.099 a gallon. How much is she going to pay at the pump
Which amount is the best estimate for the cost of the vehicle to drive through the safari in 2009?
A Chef buys 9 cucumbers, 18 peppers and 21 tomatos at the farmers market. Cucumbers ae 6 for $2, pEPPERS ARE 12 for $9 and tomatoes are 6 for $4
At a point on the ground 24ft from the base of a tree, the distance to the top of the tree is 4ft more than 3times the height of the tree. find the height of the tree.
Two parallel lines are crossed by a transversal. What is the value of g? g = 75 g = 80 g = 100 g = 105
The angle measure of g is ∠g = 105°.
The correct option is D.
What is an angle measure?When two lines or rays intersect at a single point, an angle is created. The vertex is the term for the shared point. An angle measure in geometry is the length of the angle created by two rays or arms meeting at a common vertex.
As per the information provided, it is given that the two parallel lines u and v are traversed by line g.
To find the angle measure of ∠g:
The angle measure of ∠g is equal to 105° by the alternate exterior angle theorem.
Therefore, ∠g = 105°.
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The complete question:
Two parallel lines are crossed by a transversal. What is the value of g? g = 75 g = 80 g = 100 g = 105
FOR 30 POINTS WILL MARK BRAINLIEST you have 500 sheets of notebook paper after one week you have 72% of the sheets left. you use the same number of sheets each week write an equation that represents the number y of pages remaining after 8 weeks
By graphing the system of constraints find the values of x and y that maximize the objective function, find the maximum value.
3x+y<=7
x+2y<=9
x>=0
y>=0
Maximum for P=2x+y
A. P=3
B. P=4
C. P=6
D. P=10
Answer:
x=1, y=4
P=6
Step-by-step explanation:
We are given that
[tex]3x+y\leq 7[/tex]
[tex]x+2y\leq 9[/tex]
[tex]x\geq 0[/tex]
[tex]y\geq 0[/tex]
Objective function
[tex]P=2x+y[/tex]
We have to values of x and y that maximize the P and maximum value of P .
First we change inequality equation into equality equation
[tex]3x+y=7[/tex] (I equation )
[tex]x+2y=9[/tex] (II equation)
Equation I multiply by 2 and then subtract from II equation
[tex]-5x=-5[/tex]
[tex]x=1[/tex]
Substitute x=1 in equation I
Then, we get
[tex]3(1)+y=7[/tex]
[tex]3+y=7[/tex]
[tex]y=7-3=4[/tex]
The two equation intersect at point (1,4).
Substitute x=0 in equation I
Then , we get y=7
Substitute y=0 then
[tex]3x=7[/tex]
[tex]x=\frac{7}{3}=2.3[/tex]
The equation I cut the x- axis at point (2.3,0)and y-axis at (0,7).
Substitute x=0 in equtaion II
[tex]2y=9[/tex]
[tex]y=\frac{9}{2}=4.5[/tex]
Substitute y=0
Then, we get
[tex]x=9[/tex]
Therefore, the equation II cut the x- axis at point (9,0) and y axis at point (0,4.5).
Substitute x=0 and y=0 in inequality Equation I
[tex]3(0)+0=0< 7[/tex]
It is true . Therefore, shaded region below the line.
Substitute x=0 and y=0 in inequality equation II
Then, [tex]0+2(0)=0 < 9[/tex]
It is true. Therefore, the shaded region below the line.
The feasible region is bounded.The feasible region bounded by (0,0),(0,4.5),(2.3,0) and (1,4).
At (0,4.5)
[tex]P=2(0)+4.5=4.5[/tex]
At (2.3,0)
[tex]P=2(2.3)+0=4.6[/tex]
At (1,4)
[tex]P=2(1)+4=6[/tex]
At (0,0)
[tex]P=2(0)+0=0[/tex]
Hence, maximum value of P is 6 at (1,4).
x=1 and y=4
what is the constant rate of change of y=7-13x
The constant rate of change in the linear equation y=7-13x is -13, representing the slope of the line which remains constant for any value of x. This is different from nonlinear functions, where the rate can vary with x.
Explanation:The constant rate of change in the equation y=7-13x refers to the slope of the line represented by this linear equation. In this case, the constant rate of change, or the slope, is -13. This means for every one unit increase in x, y decreases by 13 units. Since the equation is linear, this rate does not change regardless of the value of x, unlike in nonlinear functions where the rate of change can vary with x.
To elaborate, if we have a nonlinear function, the rate of change depends on the interval of x we are looking at. For a linear function like this one, the slope is the same over any interval, so there's no need to compute it over smaller ranges of x to understand the rate of change.
Moreover, in calculus, the derivative represents the rate of change, and for a linear function, the derivative is simply the coefficient of x. Hence, the derivative of y with respect to x is -13. The derivative is zero for any constant term in a function, such as the 7 in the given equation.
Suppose the weights of farmer carl's potatoes are normally distributed with a mean of 8.0 ounces and a standard deviation of 1.1 ounces. suppose carl bags his potatoes in randomly selected groups of 6. what percentage of these bags should have a mean potato weight between 7.5 and 8.5 ounces?
2. Pat is six years older than twice his Cousin Zach’s age. The sum of their ages is less than 36.
What is the greatest age that Zach could be?
Solve for x: |2x + 12| = 18
3m + 7/2 = −2m + 5/2 the / means there fractions
3m +7/2 = -2m +5/2
subtract 5/2 from each side
3m +2/2 = -2m
2/2 = 1
3m+1 =-2m
subtract 3m from each side
1=-5m
divide both sides by -5
m = 1/-5 = -1/5
m = -1/5
Carla bought a shirt for $14 and a jacket for $69. She gave the cashier $100.
How much change should Carla get back?