How do you complete the square using fractions
Completing the square with fractions involves transforming the quadratic equation into a perfect square trinomial by adding the square of half the coefficient of x to both sides. This enables solving the equation more easily by then taking the square root of both sides and isolating x.
Completing the square using fractions involves a few steps tailored to work with fractional coefficients. To make it understandable, let's explain the process step by step:
Start with the quadratic equation and ensure it is in the form ax2 + bx + c = 0.Divide all terms by 'a' (the coefficient of x2) if 'a' is not equal to 1, to make the coefficient of x2 equal to 1.Rearrange the equation so that the constant 'c' is on the other side of the equation.Take half of the coefficient of x, which is now 'b/a', and square it. This value is added both sides of the equation to form a perfect square on one side.Rewrite the left side of the equation as a squared binomial.Finally, solve for x by taking the square root of both sides and then add or subtract the constant term.For example, let's complete the square for the equation x2 + (3/2)x = 4. We take half of the coefficient of x, (3/2)/2 or 3/4, and square it to get 9/16. Adding 9/16 to both sides gives us (x + 3/4)2 = 4 + 9/16. Simplify the right side to get a single fraction, and then proceed to solve for x. Additionally, in some scenarios, we might need to multiply both the numerator and denominator by a skillfully chosen factor, such as 1/2, to facilitate simplifying or cancelling out terms.
I I'm not good with math
a) total cost = 499 + 49.99x
b) x = 5 games
total cost = 499 + 49.99(5) = 499 + 249.95 = 748.95
Alyosha and Ivan are standing between two buildings they know to be equal height. The buildings are 500 feet apart. Looking up at the westernmost building, they form a 30 degree angle. Looking to the northeastern building, they form a 45 degree angle.
A) Ivan guesses they are halfway between the buildings. Why is this a bad guess?
B) How far away from each building are they?
A rectangular vegetable garden will have a width that is 3 feet less than the length, and an area of 54 square feet. if x represents the length, then the length can be found by solving the equation: x(x−3)=54x(x-3)=54 what is the length, x, of the garden? the length is _____ feet.
Answer
Length of rectangular garden = x feet
As, Width is 3 feet less than Length.
Width = (x-3 ) feet
Area of Rectangle = Length × Breadth
Area = 54 square feet
→54 = x × (x-3)
→ x²-3 x-54=0
Splitting the Middle term
→x² - 9 x + 6 x- 54=0
→x × (x-9)+6 × (x-9)=0
→ (x+6)(x-9)=0
→x+6=0 ∧ x-9=0
x≠-6, as length can't be negative.
So, x=9
Length of Rectangular garden = 9 feet
A data set has the following characteristics: Mean: 4.9 Median: 6 Mode: 6 Variance: 4 The z-score is the number of
Answer:
Standard deviations and mean
Step-by-step explanation:
Answer:
I.
standard deviations
mean
II.
-1.95
0.05
0.8
Step-by-step explanation:
I.
The z-score is the number of standard deviations a data value is away from the mean.
II.
z1 = -1.95
z5 = 0.05
z6.5 = 0.8
A quadratic equation is shown below:
9x2 − 16x + 60 = 0
Describe the solution(s) to the equation by just determining the radicand. Show your work.
: Solve 4x2 + 8x − 5 = 0 by using an appropriate method. Show the steps of your work, and explain why you chose the method used.
hello :
help :
the discriminat of each quadratic equation :
ax²+bx+c=0 ....(a ≠ 0) is :
Δ = b² -4ac
1 ) Δ > 0 the equation has two reals solutions : x =
(-b±√Δ)/2a
2 ) Δ = 0 : one solution : x = -b/2a
3 ) Δ < 0 : no reals solutions
How many ways can we put 7 adults and 3 kids in line (where order matters) so that no two kids are next to each other?
Determine whether the cost for ordering multiple items that will be delivered is sometimes, always, or never proportional. Explain your reasoning.
Evaluate f(x)= -x^2+1 for x=-3
A.4
B.-4
C.-8
D.-9
How many cubes with 5 inches side Will completely fill a cube that is 10 inches on each side
Suppose triangle ABC has vertices at A(1, 0), B(10, 0), and C(2, 6). After a 60° counterclockwise rotation about the origin, vertex B' has coordinates (5, ?).
8/3 , 2.28, 10/12 , 0.199 what number in the list above has the greatest value?
Answer:
[tex]\frac{8}{3}[/tex] is the greatest value.
Step-by-step explanation:
The given numbers are [tex]\frac{8}{3}[/tex], 2.28, [tex]\frac{10}{12}[/tex], 0.199
In this question we have to find out greatest value.
So, first we convert all the values in decimals.
To convert [tex]\frac{8}{3}[/tex] in decimal form, we divide 8 by 3. The answer would be 2.67
2.28
[tex]\frac{10}{12}[/tex] = 0.83
0.199
Now we arrange these numbers in the increasing order.
0.199 < 0.83 < 2.28 < 2.67
So the greatest number is 2.67 that is [tex]\frac{8}{3}[/tex].
For the function f(x) = –2(x + 3)2 − 1, identify the vertex, domain, and range. The vertex is (3, –1), the domain is all real numbers, and the range is y ≥ –1. The vertex is (3, –1), the domain is all real numbers, and the range is y ≤ –1. The vertex is (–3, –1), the domain is all real numbers, and the range is y ≤ –1. The vertex is (–3, –1), the domain is all real numbers, and the range is y ≥ –1.
we have
[tex]f(x)=-2(x+3)^{2}-1[/tex]
we know that
the equation of a vertical parabola in vertex form is equal to
[tex]y=a(x-h)^{2}+k[/tex]
where
[tex](h,k)[/tex] is the vertex
If [tex]a > 0[/tex] ------> then the parabola open upward (vertex is a minimum)
If [tex]a < 0[/tex] ------> then the parabola open downward (vertex is a maximum)
In this problem
the vertex is the point [tex](-3,-1)[/tex]
[tex]a=-2[/tex]
so
[tex]-2 < 0[/tex] ------> then the parabola open downward (vertex is a maximum)
The domain is the interval-------> (-∞,∞)
that means------> all real numbers
The range is the interval--------> (-∞, -1]
[tex]y\leq-1[/tex]
that means
all real numbers less than or equal to [tex]-1[/tex]
therefore
the answer is
a) the vertex is the point [tex](-3,-1)[/tex]
b) the domain is all real numbers
c) the range is [tex]y\leq-1[/tex]
see the attached figure to better understand the problem
Rewrite the rational exponent as a radical by extending the properties of integer exponents.
Evaluate the function rule for the given value. Y=6*4^x for x=-3
Answer:
[tex]y = \frac{3}{32}[/tex]
Step-by-step explanation:
Using the exponent rule:
[tex]a^{-n} = \frac{1}{a^n}[/tex]
Given the function:
[tex]y = 6 \cdot (4)^x[/tex] ......[1]
To evaluate the function rule for x = -3.
Substitute x = -3 in [1] we have;
[tex]y = 6 \cdot (4)^{-3}[/tex]
⇒[tex]y = 6 \cdot \frac{1}{4^3} = 6 \cdot \frac{1}{64}=\frac{3}{32}[/tex]
Therefore, the function rule for the given value when x = -3 is:
[tex]y = \frac{3}{32}[/tex]
Step by step on how to solve that equation
4x-5=4x+10 solve for x
The equation 4x - 5 = 4x + 10 has no solution because subtracting 4x from both sides yields -5 = 10, which is a contradiction.
Explanation:The equation 4x - 5 = 4x + 10 cannot be solved for x in the usual way because attempting to isolate x on one side will result in a contradiction. If we subtract 4x from both sides of the equation, we get -5 = 10, which is not true for any value of x. Therefore, this equation has no solution.
Anthony was tracking the increasing number of animals in the zoo. at 9
a.m., there were 56 animals. at 11
a.m., there were 60 animals. if anthony made the function f(x) = 2x − 38, what would the 2 represent? the number of animals at midnight the rate at which the number of animals was increasing the length of time he recorded for the total change in the number of animals
It’s a little surprising that this question didn’t come up earlier. Unfortunately, there’s no intuitive way to understand why “the energy of the rest mass of an object is equal to the rest mass times the speed of light squared” (E=MC2). A complete derivation/proof includes a fair chunk of math (in the second half of this post), a decent understanding of relativity, and (most important) experimental verification.
Answer:
2 would represent the rate at which the animals were increasing
Step-by-step explanation: This all comes back to the slope intercept form:
y = mx + b
2x is the slope, and the slope is the rate of which a point will increase or decrease.
Find the 5th term in the expansion of (x – 3y)8
A soccer team has won 1/2 of the games they played. they won 12 games. how many games did they play? Show your working/calculations
they won 1/2 of their games and won 12 games
1/2 = 50% , multiply by 2 to make 100%
so 12 x 2 = 24 games were played
To calculate the number of games played by the soccer team, multiply the number of games they won (12) by 2, because winning 12 games represents winning [tex]\frac{1}{2}[/tex] of all games played. The soccer team thus played a total of 24 games.
To find how many games the soccer team played in total, we can set up a proportion using the information that they won [tex]\frac{1}{2}[/tex] of their games. This means for every game they won, they played two (since [tex]\frac{1}{2}[/tex] represents one out of two games). If the team won 12 games, and that represents [tex]\frac{1}{2}[/tex] of the total games, we can use a simple equation to solve for the total number of games (T).
The proportion is as follows: [tex]\frac{1}{2}[/tex] = [tex]\frac{12}{T}[/tex], where 12 is the number of games won, and T is the total number of games played.
To solve for T, we multiply both sides of the equation by T to get rid of the fraction: [tex]\frac{T}{2}[/tex] = 12. To find T, we then multiply both sides of the equation by 2 to isolate T on one side of the equation: T = 12 * 2.
Therefore, the team played a total of T = 24 games.
A Sporting good store charges $30 for 12 cans of tennis balls. The tennis coach orders 100 cans of tennis balls for the tennis team. How much were the coach pay for the tennis balls?
Use Gauss-Jordan elimination to solve the following linear system.
x – 2z = 9
6x – 2y – 5z = 29
–5x + 5y + 3z = –14
A. (–5,–3,0)
B. (3,2,–3)
C. (5,3,–5)
D. (3,6,–4)
By using Gauss-Jordan elimination the solution to the linear equation is (3, 2, –3). The correct option is B.
What is Gauss-Jordan's elimination method?
In mathematics, the algorithm known as Gaussian elimination, commonly referred to as row reduction, is used to solve systems of linear equations. It comprises a series of operations carried out on the relevant coefficients matrix.
x – 2z = 9 so x = 2z + 9
6x – 2y – 5z = 29
–5x + 5y + 3z = –14
Substitute x = 2z + 9 into 6x – 2y – 5z = 29 and –5x + 5y + 3z = –14
6(2z + 9) – 2y – 5z = 29
12z + 54 - 2y -5z =29
-2y + 7z = - 25 (1st equation)
–5(2z + 9) + 5y + 3z = –14
-10z - 45 + 5y + 3z = -14
5y - 7z = 31 (2nd equation)
Multiply (1st) equation by 5 and (2nd) equation by 2.
-10y + 35z = - 125
10y - 14z = 62
21z = - 63
z = -3
Substitute z = - 3 into x = 2z + 9
x = 2z + 9
x = 2(-3) + 9
x = 3
Substitute x = 3 and z = 3 into 6x – 2y – 5z = 29
6x – 2y – 5z = 29
6(3) – 2y – 5(-3) = 29
18 - 2y + 15 = 29
-2y + 33 = 29
-2y = -4
y = 2
x = 3, y = 2 and z = -3
Therefore, the solution to the linear equation is (3, 2, –3). The correct option is B.
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What do the parallel lines shown on segment BD and segment DC represent?
it means both sections are equal.
so BD = 18 and DC = 18
Kris wanted to understand whether studentstudents at her school were in favor of an extended school day. She surveyed some students and displayed the results in the table below:
In favor
Opposed
Undecided
Grade 9
6
4
8
Grade 10
10
11
9
Grade 11
12
15
11
Grade 12
15
6
14
If the principal randomly selects a student in grade 10 from this survey, what is the probability that the student is opposed to extending the school day?
Answer:
The probability that the student is opposed to extending the school day is 0.3666 or 36.67% approx.
Step-by-step explanation:
Arranging the table properly and adding a column of total students.
In favor Opposed Undecided Total students
Grade 9 6 4 8 18
Grade 10 10 11 9 30
Grade 11 12 15 11 38
Grade 12 15 6 14 35
If the principal randomly selects a student in grade 10 from this survey, this means we will only consider grade 10 students.
The probability that the student is opposed to extending the school day is = [tex]\frac{number of opposed students}{total number of students}[/tex]
There are 11 students who are opposed and total students in grade 10 are 30.
So, probability is : [tex]\frac{11}{30}= 0.3666[/tex] or 36.67% approx.
We generally report a measurement by recording all of the certain digits plus ______ uncertain digit(s).
We generally report a measurement by recording all of the certain digits plus one uncertain digit.
What are significant figures?In positional nomenclature, a number's real numbers are its dependable and essential digits for indicating how much of something there is.
If a measurement's result is expressed by a number with more digits than the measurement resolution permits, only those digits up to the measuring resolution's maximum are trustworthy, and only those digits could be significant figures.
Typically, we record all the confirmed digits of measurement together with one questionable digit.
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HELP ROUND 604703.472883 TO THE NEAREST HUNDRED
Answer:
The required number is 604703
Step-by-step explanation:
Consider the provided number 604703.472883
The rule for rounding a number is:
If the number on the right side of rounding digit is 0, 1, 2, 3, 4 then no need to change the rounding digit and change the rest of the digit right to rounding digit with 0.
If the number on the right side of rounding digit is 5, 6, 7, 8, 9 then rounding digit rounds up by one number and change the rest of the digit right to rounding digit with 0.
The number on the hundred place is 7 and the number on right side is 0.
Thus, no need to change the rounding digit and change the rest of the digit right to rounding digit with 0.
Hence, the required number is 604703
Which sequence is geometric and has 1/4 as its fifth term and 1/2 as the common ratio?
A certain bag of potting soil is 1/4 peat moss,and the rest is dirt. What part is dirt?
The part of the bag that consist of dirt is:
3/4
Step-by-step explanation:It is given that the amount of peat moss that is present in a bag of potting soil is:
1/4
Hence, the remaining space that is left in the bag consist of dirt.
Remaining space is calculated as :
[tex]1-\dfrac{1}{4}\\\\\\=\dfrac{3}{4}[/tex]
( Since we subtract 1/4 from a whole of a bag)
Hence, the part that is dirt in the bag of potting soil is:
3/4
A line with a slope of 4 passes through (6, 11). Which choice is an equation of this line?
Answer:PLEASE MARK AS BRAINIEST
y – 11 = 4(x – 6)
How to determine if a series has a sum?