We must solve the following quadratic equation by completing squares:
[tex]x^2+5x+6=0.[/tex]1) First, we rewrite the equation in the following way:
[tex]x^2+2\cdot\frac{5}{2}\cdot x+6=0.[/tex]2) We sum to both sides of the equation (5/2)², get:
[tex]x^2+2\cdot\frac{5}{2}\cdot x+(\frac{5}{2})^2+6=(\frac{5}{2})^2.[/tex]3) Expressing the first term as a square, we have:
[tex](x+\frac{5}{2})^2+6=\frac{25}{4}.[/tex]4) Passing the +6 at the left as -6 at the right:
[tex]\begin{gathered} (x+\frac{5}{2})^2=\frac{25}{4}-6, \\ (x+\frac{5}{2})^2=\frac{25}{4}-\frac{24}{4}, \\ (x+\frac{5}{2})^2=\frac{1}{4}. \end{gathered}[/tex]5) By taking the square root on both sides, we get two solutions:
[tex]\begin{gathered} x+\frac{5}{2}=+\frac{1}{2}\rightarrow x=\frac{1}{2}-\frac{5}{2}=-\frac{4}{2}=-2, \\ x+\frac{5}{2}=-\frac{1}{2}\rightarrow x=-\frac{1}{2}-\frac{5}{2}=-\frac{6}{2}=-3. \end{gathered}[/tex]Answer1) Re-writing the equation:
[tex]x^2+2\cdot\frac{5}{2}\cdot x+6=0.[/tex]2) Summing (5/2)² on both sides:
[tex]x^2+2\cdot\frac{5}{2}\cdot x+(\frac{5}{2})^2+6=(\frac{5}{2})^2.[/tex]3) Expressing the first term as a square:
[tex](x+\frac{5}{2})^2+6=\frac{25}{4}.[/tex]4) Simplifying:
[tex]\begin{gathered} (x+\frac{5}{2})^2=\frac{25}{4}-6, \\ (x+\frac{5}{2})^2=\frac{25}{4}-\frac{24}{4}, \\ (x+\frac{5}{2})^2=\frac{1}{4}. \end{gathered}[/tex]5) Taking the square root:
[tex]\begin{gathered} x+\frac{5}{2}=+\frac{1}{2}\rightarrow x=\frac{1}{2}-\frac{5}{2}=-\frac{4}{2}=-2, \\ x+\frac{5}{2}=-\frac{1}{2}\rightarrow x=-\frac{1}{2}-\frac{5}{2}=-\frac{6}{2}=-3. \end{gathered}[/tex]Convert each slope-intercept or point slope equation into standard form.
y - 3 = 1/5(x + 6)
The Standard form of the equation will be;
⇒ x - 5y = -21
What is Standard form of equation?
The standard form of the equation is defined as;
Ax + By = C
Where, A, B and C are integers.
Given that;
The equation in slope - intercept form as;
⇒ y - 3 = 1/5 (x + 6)
Now,
We convert the equation in standard form as;
Since, The equation in slope - intercept form as;
⇒ y - 3 = 1/5 (x + 6)
Change into standard form as;
⇒ y - 3 = 1/5 (x + 6)
Multiply by 5 both side, we get;
⇒ 5( y - 3) = (x + 6)
⇒ 5y - 15 = x + 6
Add 15 both side, we get;
⇒ 5y - 15 + 15 = x + 6 + 15
⇒ 5y = x + 21
Subtract 21 both side, we get;
⇒ 5y - 21 = x + 21 - 21
⇒ 5y - 21 = x
Subtract 5y both side, we get;
⇒ 5y - 21 - 5y = x - 5y
⇒ - 21 = x - 5y
⇒ x - 5y = -21
Therefore,
The Standard form of the equation will be;
⇒ x - 5y = -21
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Yesterday, grace drove 28 1/2 miles. She used 1 1/4 gallons of gasoline. What is the unit rate for miles per gallon?
Answer: 22.8 miles or 22 4/5
Step-by-step explanation:
To find the unit rate you do
28.5 ÷ 1.25 = 22.8
Find the largest of three consecutive odd integers whose sum is 111.
The largest of three consecutive odd integers whose sum is 111 is 39
What is an integer?Positive, negative, and zero are all examples of integers. The Latin word "integer" signifies "whole" or "intact." As a result, fractions and decimals are not included in integers.
Odd integers that follow each other grow (or shrink) by a factor of 2. Consider the numbers 1, 3, and 5. Add two to the preceding number to move from one to the following. You don't know where to begin, and that is the issue here. In actuality, you are searching for the least of the three integers, therefore this is your unknown.
x + (x + 2) + (x + 4) = 111
x + x + 2 + x + 4 = 111
3x + 6 = 111
3x = 105
x =[tex]\frac{105}{3}[/tex]
x = 35
35 ,37,39
The largest of three consecutive odd integers whose sum is 111 is 39
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what is the slope of the line that passes through the points (-7,-8) and (-5,-6)? Write your answer in simplest form.
Answer:
m=1
Step-by-step explanation:
Point P(-3, 4) is a point on the terminal side of 0 in standard form. Find the exact value ofsine, cosine, and tangent for 0.
Solution
Step 1
The terminal side, containing point (-3, 4) is located in Quadrant 2.
sine is positive
cosine and tangent are both negative.
Step 2
Draw a diagram to illustrate the information
Step 3
[tex]\begin{gathered} Find\text{ d using the Pythagoras theorem} \\ d^2\text{ = 3}^2+\text{ 4}^2 \\ d^2\text{ = 9 + 16} \\ d^2\text{ = 25} \\ d\text{ = }\sqrt{25} \\ \text{d = 5} \end{gathered}[/tex]Step 4:
[tex]\begin{gathered} sine\text{ = }\frac{Opposite}{Hypotenuse} \\ Sin\theta\text{ = }\frac{4}{5} \end{gathered}[/tex][tex]\begin{gathered} Cosine\text{ = }\frac{Adjacent}{Hypotenuse}\text{ = }\frac{x}{d} \\ Cos\theta\text{ = }\frac{-3}{5} \end{gathered}[/tex][tex]\begin{gathered} tangent\text{ = }\frac{Opposite}{Adjacent}\text{ = }\frac{y}{x} \\ tan\theta\text{ = }\frac{-4}{3} \end{gathered}[/tex]Final answer
[tex]\begin{gathered} sin\theta\text{ = }\frac{4}{5} \\ cos\theta\text{ = }\frac{-3}{5} \\ tan\theta=\frac{-4}{3} \end{gathered}[/tex]Two trains leave the same station at the same time, one traveling west at a constant speed of 60 miles per hour, the other traveling south at a constant speed of 80 miles per hour. After how long are the two trains exactly 300 miles apart?
After 2.14 hours the trains are exactly 300 miles apart.
How to find the time the trains travel 300 miles apart?Two trains leave the same station at the same time, one traveling west at a constant speed of 60 miles per hour, the other traveling south at a constant speed of 80 miles per hour.
The time both of them will travel 300 miles can be calculated as follows:
Therefore,
speed = distance / time
distance = speed × time
The train that travel west:
let
t = time of the train that travel west
distance = 60t
The train that travel south:
distance = 80t
Therefore,
total distance = 60t + 80t
300 = 140t
t = 300 / 140
t = 2.14285714286
t = 2.14 hours
Therefore,
time taken = 2.14 hours
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Evaluate the expression below using the properties of operations. -36 ÷ 1/4 • (-1/8) • (-3) ÷ 6
Answer:
-9
Step-by-step explanation:
We are working with multiplications and divisions, which have no precedence over each other. So we can do the operations in order. It is important to remind that in these operations, if two numbers have different signals, the result of the operation is negative, otherwise, positive.
-36 ÷ 1/4
Different signals, so the result will be negative
[tex]\frac{-36}{\frac{1}{4}}=-36\ast\frac{4}{1}=-36\ast4=-144[/tex]-36 ÷ 1/4 • (-1/8) = -144 * (-1/8)
Same signal, so positive
[tex]-144\ast(-\frac{1}{8})=\frac{144}{8}=18[/tex]-36 ÷ 1/4 • (-1/8) • (-3) = 18 * (-3) = -54
-36 ÷ 1/4 • (-1/8) • (-3) ÷ 6 = -54 ÷ 6 = -9
Whats a ratio for 5ml and 120ml
Step 1:
The ratio for 5ml to 120ml
Step 2:
The symbol of ratio is :
Therefore,
[tex]\begin{gathered} 5ml\text{ ratio 120ml} \\ =\text{ 5 : 120} \\ =\text{ }\frac{5}{120} \\ =\text{ }\frac{1}{24} \\ =\text{ 1 : }24 \end{gathered}[/tex]Step 3
Final answer
1 : 24
Kali just started a new sales floor job to save for college. She earns 15.75 plus a flat fee of 50 . She wants to earn between 200 and 400 . The following inequality represents her earning potential
200 ≤ 15.75x + 50 ≤ 400 Solve the inequality PLEASE HELP ASAP
!!
The given inequality has the following solution set
9.52 ≤ x ≤ 22.22
If we express this as an interval, we get [9.52, 22.22].
Here is the inequality which is Kali's earning potential
200 ≤ 15.75x + 50 ≤ 400
To solve the inequality, we must isolate the variable in the center; if we remove 50 from each of the three sides, we get:
200 - 50 ≤ 15.75x + 50 - 50 ≤ 400 - 50
150 ≤ 15.75x ≤ 350
Now we must divide both totals by 15.75, yielding:
150/15.75 ≤ 15.75x/15.75 ≤ 350/15.75
9.52 ≤ x ≤ 22.22
This is the inequality's solution; the solution set expressed as an interval will be [9.52, 22] or 9.52 ≤ x ≤ 22.22
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8. A car costs $10,500, and you're offered a loan that requires $800 down and a monthly payment of $187.53 for 60 months, how much will you pay in interest? Round your answer to the nearest dollar.$
The Solution:
Given that a car that cost $10500 was offered as a loan with a down payment of $800.
This means the loan balance will now be:
[tex]\text{Loan baleance=10500-800= \$9700}[/tex]The loan payment plan is a monthly payment of $187.53 for 60 months.
[tex]\text{Total Payment=187.53}\times60=\text{ \$11251.80}[/tex]We are required to find how much was paid in interest.
We shall take the difference between the total payment and the loan balance.
[tex]\begin{gathered} \text{Interest paid=Total payment-Loan balance} \\ \text{Interest paid=11251.80-9700= \$1551.80}\approx\text{ \$1552} \end{gathered}[/tex]Therefore, the correct answer is $1552
How long do people typically spend traveling to work? The answer may depend on where they live. Here are the travel times in minutes of 20 randomly chosen workers in New York state
Based on the information given, the standard deviation for the observation is: 23.802698 or approximately 24.
The standard deviation in statistics is a measure of the degree of variation or dispersion in a set of values. A low standard deviation implies that the values are close to the set's mean, whereas a high standard deviation shows that the values are spread out over a larger range.
The formula for Standard deviation is given as:
σ[tex]= \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2} .[/tex]
Where
σ = population standard deviation
[tex]\bar x[/tex] = mean
x = each value of the population
n = number of observation
Note that mean ([tex]\bar x[/tex]) = [tex]\( \frac{1}{n} \sum_{i=i}^{n} x_{i} \),[/tex]
[tex]\bar x[/tex] = (5+10+10+10+10+12+15+20+20+25+30+30+40+40+60+60+65+70+70+70)/20
= 33.6
Hence the standard deviation for travel times for these 20 New York workers is:
Standard Deviation = [tex]\sqrt{\frac{\sum(x_{1} - {\bar x})^{2} }{n-1} }[/tex]
= √ [(5-33.6)²+ (10 - 33.6)² + .... + (70 - 33.6)²]/(20-1)]
Standard Deviation = √(10764.8)/(20-1)
SD = √(10764.8/19)
SD = √566.56842
SD = 23.802698
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Full Question:
How long do people spend traveling to work? The answer may depend on where they live here are the travel times in minutes for 20 workers in New York chosen at random by the census bureau:
5 10 10 10 10 12 15 20 20 25 30 30 40 40 60 60 65 70 70 70
What is the standard deviation for travel time for these 20 New York state workers?
One month Tony rented 5 movies and 3 video games for a total of $32. The next month he rented 2 movies and 12 video games for a total of $83. Find therental cost for each movie and each video game.Rental cost for each movie:Rental cost for each video game:
Let x represent the rental cost for each movie.
Let y represent the rental cost for each video game.
We were told that One month Tony rented 5 movies and 3 video games for a total of 532. This means that
5x + 3y = 32
Also, the month, he rented 2 movies and 12 video games for a total of $83. This means that
2x + 12y = 83
Dividing through by 2, we have
x + 6y = 41.5
x = 41.5 - 6y
Substituting x = 41.5 - 6y into 5x + 3y = 532, we have
5(41.5 - 6y) + 3y = 32
207.5 - 30y + 3y = 32
- 30y + 3y = 32 - 207.5
- 27y = - 175.5
y = - 175.5/- 27
y = 6.5
x = 41.5 - 6(6.5) = 41.5 - 39
x = 2.5
Rental cost for each movie = $2.5
Rental cost for each video game = $6.5
If z varies inversely as w, and z=20 when w=6, find z when w=3.
Z=
Need help with absolute value please
Answer:
let's say you need to find the absolute value of -5 the answer would be 5.
Step-by-step explanation:
Since the distance between -5 and 0 in a number line is 5. This would also be applied If you you were trying to find the absolute value of 5. it would also be 5.
When in math, I figured out negative number's absolute value will always be positive. absolute value numbers of positive numbers will stay the same.
Write the equation of the line with the given information in point-slope form. (-5, 11) and (-2, 1)
Answer
The equation in point-slope form is
y - 1 = (-10/3) (x + 2)
We can then simplify this by multiplying through by 3 to obtain
3y - 3 = (-10) (x + 2)
3y - 3 = 10x - 20
3y = 10x - 20 + 3
3y = 10x - 17
Explanation
The general form of the equation in point-slope form is
y - y₁ = m (x - x₁)
where
y = y-coordinate of a point on the line.
y₁ = This refers to the y-coordinate of a given point on the line
m = slope of the line.
x = x-coordinate of the point on the line whose y-coordinate is y.
x₁ = x-coordinate of the given point on the line
We can calculate the slope of the line and then use any of the two points given to serve as the point (x₁, y₁) in the equation
For a straight line, the slope of the line can be obtained when the coordinates of two points on the line are known. If the coordinates are (x₁, y₁) and (x₂, y₂), the slope is given as
[tex]Slope=m=\frac{Change\text{ in y}}{Change\text{ in x}}=\frac{y_2-y_1}{x_2-x_1}[/tex]For this question,
(x₁, y₁) and (x₂, y₂) are (-5, 11) and (-2, 1)
x₁ = -5
y₁ = 11
x₂ = -2
y₂ = 1
[tex]\text{Slope = }\frac{1-11}{-2-(-5)}=\frac{-10}{-2+5}=\frac{-10}{3}[/tex]Slope = m = (-10/3)
Using the point (-2, 1) as (x₁, y₁), we can write the equation of the line
y - y₁ = m (x - x₁)
y - 1 = (-10/3) (x - (-2))
y - 1 = (-10/3) (x + 2)
We can then simplify this by multiplying through by 3 to obtain
3y - 3 = (-10) (x + 2)
3y - 3 = 10x - 20
3y = 10x - 20 + 3
3y = 10x - 17
Hope this Helps!!!
(2tanθ-3cosθ) Expand
a=460 rounded to the nearest 10 b=11.9 rounded to 1 dp find the minimum (to 2 dp) of a / b
The minimum result (to 2 dp) of a / b is 38.66
What is rounding decimals?The term "rounding decimals" refers to the accurate rounding of decimal figures. When rounding a decimal number, certain principles must be followed. Simply put, if the last digit is less than 5, round down the previous digit. However, if it is 5 or greater, round the previous digit up.Given:
a=460 rounded to the nearest 10
b=11.9
Now, substitute the values of a and b in a/b,
a/b = 460/11.9
Multiply the same integer(10) by both the numerator and denominator,
a/b = 4600/119
Round the number obtained
a/b ≅ 38.66
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help meeeeeeeeeeeeeeeeeee pleaseeeeeeeee!!!
Answer:
I think it is y
Step-by-step explanation:
find an equation of the circle that has center (1,-5) and passes through (2,1)
1) Since the equation of the circle is given by:
[tex](x-h)^2+(y-k)^2=r^2[/tex]And we've been told the Center (1,-5) and one point located at the circumference, (2,1). So let's find the radius, i.e. the distance from the center to any point to the circumference.
2) Let's use the Formula for the distance between (1,-5) and (2,1), derived from the Pythagorean Theorem:
[tex]\begin{gathered} d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ d=\sqrt[]{(2-1)^2+(1_{}+5)^2} \\ d=\sqrt[]{37} \end{gathered}[/tex]3) So d = radius, and now we can plug those pieces of information into the formula of the circle:
[tex]\begin{gathered} (x-h)^2+(y-k)^2=r^2 \\ (x-1)^2+(y+5)^2=(\sqrt[]{37})^2 \\ (x-1)^2+(y+5)^2=37 \end{gathered}[/tex]So we now have the formula for that circle.
Answer:
Radius (R) is equal to the distance between the points (-2,1) and (3,1)
R² = (1 - 5)² + (1 - 1)² = (-24)² + 0 = 579
R = 23.16
2(y−1)+6y = −10 please help fastest answer gets 37 points
How many solutions does this system of equations have?
y = x2 + x + 3
y = -2x - 5
Answer: (x, y) = (-8/5, -9/5)
Write the quadratic function in vertex form. Then identify the vertex.
g(x)=x^2+12x+37
The vertex form is y= (x+ 6)²+1.
What is Vertex form?The vertex form of a quadratic equation is y = a (x- h)² + k as opposed to the regular quadratic form, which is an x² + bx + c = y. In both cases, the variables that indicate whether the parabola is facing up (+ a) or down ( a) are y, the y-coordinate, x, and a.
a=1
b=12
c=37
Consider the vertex form of a parabola.
a(x+ d)²+e
Now, d= b/2a
d=12/ 2
d=6
and, e= c-b²-4a
e= 37 - (12)²/4x1
e= 37 - 36
e= 1
Then, the vertex form is
y = a(x+ d)²+e
y= 1(x+ 6)²+1
y= (x+ 6)²+1
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11. Algebra The total cost of the Fatigato
family's two cars was $71,482. The cost of
one car was $38,295. Write an equation
using a variable to represent the cost of
the family's other car.
Answer:
$33,187
Step-by-step explanation:
Our total is 71,482 so let's set our equation to 71482=_________
Now we have $38,295 for one car, and we will subtract it from the total to find the other car, as there are 2 cars.
Equation:
71482-38295=x OR 71482=38295+x
X is the cost of the second car.
A circle has a diameter of 24 centimeters. Central angle FOG is drawn, determining an arc FG. The radian measure of angle FOG is 3/4 What is the length of arc FG in centimeters?16 cm9 cm32 cm18 cm
Answer:
9 cm
Explanation:
We are given the following information:
Diameter = 24 cm
Angle FOG = 3/4 rad
The formula for calculating arc length is written below:
[tex]\begin{gathered} L=\theta\times r \\ where\colon \\ \theta=central\text{ angle of }arc,\text{ in }rad \\ r=radius \\ r=\frac{\text{Diameter}}{2}=\frac{24}{2}=12cm \\ \theta=\frac{3}{4}rad \\ \text{Substitute these into the formula, we have:} \\ L=\frac{3}{4}\times12 \\ L=\frac{3\times12}{4} \\ L=9cm \\ \\ \therefore L=9cm \end{gathered}[/tex]Therefore, the arc length is 9 cm
Brittany has 34,011 in a savings account that earns 6% annually. the interest is not compounded. How much interest will she earn in 4 years? use the formula I = PRT, where I is the interest earned, p is the principal, r as the interest rate expressed as a decimal, and T is the time in years.
Notice that we are dealing with simple interest, and therefore given by the formula:
I = P * R * T
where P = $34,011
R = 6% in "decimal" form (0.06)
T = 4 (for 4 years)
Then, we have:
I = 34011 * 0.06 * 4 = $8162.64
This is the interest Brittany earned in 4 years.
please help me solve. I have the answer in yellow, but it's not correct.
Remember that
27=(3)(3)(3)=3^3
so
[tex]\sqrt[]{3\cdot}\sqrt[\square]{27}=\sqrt[]{3\cdot}\sqrt[\square]{3^3}=\sqrt[]{3^4}=3^2=9[/tex]answer is 9
so
9 NA
blank 1 ------> 9
blank 2 ------> NA
what is the value of a in the equation 5a – 10b = 45, when b =3?a)3b)15c)21d)39
SOLUTION
Given the question, the following are the solution steps to answer the question.
STEP 1: Write the given equation
[tex]5a-10b=45[/tex]STEP 2: Substitute 3 for b in the equation
[tex]5a-10(3)=45[/tex]STEP 3: Simplify the equation to solve for a
[tex]\begin{gathered} 5a-30=45 \\ 5a=45+30 \\ 5a=75 \\ a=\frac{75}{5} \\ a=15 \end{gathered}[/tex]Hence, the value of a is 15
Find X. Circumscribed Angles
The value of of angle x of the circumscribing circle is x° = 140°
In the above question, the following figure is given, where
The angle inside the circle made by the intersection of two line segments is = 40°
We need to find the angle x made by the angle made by the tangents outside the circle
A line that touches a curve or a circle at one point is said to be tangent.
The point of tangency is the intersection of the tangent line and the curve.
We'll find the value of of angle x using the theorems of the circumscribing circle.
The sum of opposite angles of a circumscribing quadrilateral is always 180°
Using this property we can write,
40° + x° = 180°
x° = 180° - 40°
x° = 140°
Hence, the value of of angle x using the theorems of the circumscribing circle is x° = 140°
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Describe the shape, orientation, and vertex of
each parabola relative to the graph of y=x².
Sketch each graph.
a) y=-0.5x² + 2
c) y = -0.1x² - 6
e) y=-3x²-5
g)y=8x²+4
b) y = 2x²
d)y=x²+4
f) y=0,1x² +2
h) y=-0.7x²-3
The parabola y = - 0.5x² + 2 opens downwards and the vertex is at (0,2) .
A parabola is a mirror-symmetric planar curve with a rough U-shape. It can be defined by many seemingly unrelated mathematical descriptions that all relate to the same curves.
One way to interpret a parabola is with a line and a point (the focus) (the directrix). The directrix is less significant. The parabola lies between the directrix or the focus and the endpoints in this plane that are uniformly spaced apart.
a) y=-0.5x² + 2 vertex is at (0,2)
c) y = -0.1x² - 6 vertex at (0,6)
d)y=x²+4 vertex at (0,-4)
e) y=-3x²-5 vertex at (0,5)
g)y=8x²+4 vertex at (0,-4)
h) y=-0.7x²-3 vertex at (0,3)
A parabola can also be thought of as a conic section formed by joining a right circular conical area with a plane perpendicular to another axis.
The graph of the parabola is attached below.
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If one of the flights is randomly selected find the probability that the flight silicon Rosa United Airlines flight given that it was on time.
Total of flights on time: 22 + 53 = 75
probability that the flight selected was silicon Rosa United Airlines:
53/75 = 0.71 = 71%