Step-by-step explanation:
So a quadratic equation in factored form is generally given in the form: [tex]a(x-b)(x+c)[/tex]. Sometimes x may have a coefficient but I'll just create a simple quadratic equation. In this form it's easy to see that the solutions to the equation are b and -c, because if you input them in as x, you'll get one of the factors equal to 0, and it'll make the y-value zero, thus it's a solution. So b and c can literally be anything. But for this example I'll chose 3 and 6 to keep it simple. Inputting these as b and c you'll get: [tex]a(x-3)(x+6)[/tex]. Now the value of a will determine the stretch/compression, but to keep it really simple I'll just say that a=1. So now all you have to do is expand the two factors. When you expand them you'll get it in the form: [tex](x-b)(x+c) = x^2+cx-bx-bc[/tex] by using the foil method. So if you expand the two factors in this example you'll get: [tex]x^2+6x-3x-18[/tex] which simplifies to [tex]x^2+3x-18[/tex]. So cool now we have our equation. Now it's time to solve it, you can either factor it (since you literally just had it in factored form), or use the quadratic formula I'll show both methods
Factoring:
When you factor an equation in the form: [tex]ax^2+bx+c[/tex]. You usually multiply the coefficients a and c. Then you'll find factors of ac that add up to b. In this example ac is really just c because a is 1...
So given the equation: [tex]x^2+3x-18[/tex] You want to look for factors of -18 that add up to 3. And whenever you have a negative number as AC you really want to look for factors that have a distance of b which in this case is 3. For example I would say that 9 and 3 have a distance of 6. And then after that depending on whether b is negative or positive. I can choose which factor will be negative and which will be positive. If b was 6 well then 9 would have to be positive and 3 would be negative, but if b was -6 then 9 would have to be negative and 3 would have to be positive.
So in this example look for factors that have "distance" of 3. Well 6 and 3 have a "distance" of 6 so I know those 2 will be the factors. Since 3 is positive that means 6 will be positive, and 3 will be negative. so they add up to 3 and multiply to get -18
AC factors that add up to 3 and multiply to -18: 6 and -3
Now write it in factored form using the two factors:
[tex](x-3)(x+6)[/tex]
Now set up each factor equal to 0 and solve for x:
[tex]x-3 = 0\\x=3[/tex]
[tex]x+6=0\\x=-6[/tex]
This gives you the two solutions: x=3 and x=-6
Quadratic Formula:
The quadratic formula is usually used when a equation can't be easily factored. The quadratic formula gives both zeroes in the equation: [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]. Notice the plus or minus sign next to the square root? That's where the the two solutions will come from, or possibly one solution if it's at the vertex. or maybe even no real solutions if the quadratic never intersects the x-axis.
Ok so let's identify a, b, and c. We're given the equation: [tex]x^2+3x-18[/tex]. so a=1, b=3, and c=-18. Plugging these values into the equation you get:
[tex]x=\frac{-(3)\pm\sqrt{(3)^2-4(1)(-18)}}{2(1)}\\[/tex]
Simplifying it a bit gets you the equation:
[tex]x=\frac{-3\pm\sqrt{81}}{2}[/tex]
Simplifying the radical gives you:
[tex]x=\frac{-3\pm9}{2}[/tex]
Use the positive sign:
[tex]x=\frac{-3+9}{2}\\x=\frac{6}{2}\\x=3[/tex]
Use the negative sign:
[tex]x=\frac{-3-9}{2}\\x=\frac{-12}{2}\\x=-6[/tex]
This gives you the two solutions x=3, and x=-6
Triangle R Q S is cut by line segment T U. Line segment T U goes from side Q R to side Q S. The length of Q T is 32, the length of T R is 36, the length of Q U is 40, and the length of U S is 45.
Use the converse of the side-splitter theorem to determine if T R is parallel to R S. Which statement is true?
Line segment TU is parallel to line segment RS because StartFraction 32 Over 36 EndFraction = StartFraction 40 Over 45 EndFraction.
Line segment TU is not parallel to line segment RS because StartFraction 32 Over 36 EndFraction not-equals StartFraction 40 Over 45 EndFraction.
Line segment TU is parallel to line segment RS because StartFraction 32 Over 45 EndFraction = StartFraction 40 Over 36 EndFraction.
Line segment TU is not parallel to line segment RS because StartFraction 32 Over 45 EndFraction not-equals StartFraction 40 Over 36 EndFraction.
The statement that is true is option A that is Line segment TU is parallel to line segment RS because StartFraction 32 Over 36 EndFraction = StartFraction 40 Over 45 EndFraction.
What is the line segment about?The law of the side- splitter connote that when f the line is parallel to a side of the triangle and if that said line do intersects the other 2 sides, tt is said to divides those given sides proportionally.
So the converse means that when the sides are proportional so therefore, the side TU is said to be parallel to the side RS.
Thus;
[tex]\frac{QT}{TR}[/tex] = [tex]\frac{32}{36} = \frac{8}{9}\\\\ \\[/tex]
[tex]\\\frac{QU}{US} =\frac{40}{45} = \frac{8}{9}\\[/tex]
Therefore, the ratios are equal and as such TU is parallel to RS
Hence, The statement that is true is option A that is Line segment TU is parallel to line segment RS because StartFraction 32 Over 36 EndFraction = StartFraction 40 Over 45 EndFraction.
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Answer: OPTION A
Step-by-step explanation:
The expression x-y equals 0.7 if x and y have certain values. Evaluate the following expression with the same values of x and y
1/x-y
Answer:
1/0.7
Step-by-step explanation:
The expression x-y equals to 0.7 so we have to substitute 0.7 in the place of x-y....
----> 1/0.7
Dividing would make this equal to a long division answer (1.42857142857) so let's just leave it as a fraction!
I hope this helps! :)
Can you help me with these two because I don't understand
Answer: 1: 34 units 2: 24.5
Step-by-step explanation:
For figure 1, just split the area into squares and triangles and find the area of each. If you split figure 1 into 1 square and 3 triangles you get...
4x4= 16
(4x3)/2= 6
(4x4)/2= 8
(4x2)/2= 3
16+6+8+3 = 34
You do the same for figure 2, but use 4 triangles and 1 tiny square
1x1= 1
(4x3)/2= 6
(3x3)/2= 4.5
(4x4)/2= 8
(5x5)/2= 5
1+6+4.5+8+5= 24.5
You shouldn't need to worry about negative numbers because you are looking at the area
Show that √5×√20 is an integer?
Answer:
See below.
Step-by-step explanation:
1) Simplify √20 to 2√5.
√5 × 2√5
2) Simplify.
2√5×5
3) Simplify 5 × 5 to 25.
2√25
4) Since 5 × 5 = 25, the square root of 25 is 5.
2 × 5
5) Simplify.
10
When evaluated, this expression equals a positive whole number, therefore showing it is a integer.
What are two angles next to each other called?
Two angles which are next to each other is referred to as adjacent angle in this case.
What is an Adjacent angle?These are angles which have a common vertex and side and do not overlap with one another.
This angle lie next to each other when plotted hence the reason why they share a common vertex.
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1. Use the graph to predict the number of CDs the students at your high school bought in 2008.
The number of CDs the students at your high school bought in 2008 is 100
How to predict the number of CDs?The graph of the question is added as an attachment
From the graph, we have the following highlights:
The graph reaches a maximum of 200 in 2004Then it decreases by 25 each year2008 is 4 years from 2004.
So, the prediction is;
Year 2008 = 200 - 4 * 25
Evaluate
Year 2008 = 100
Hence, the number of CDs the students at your high school bought in 2008 is 100
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please help ill appreciate it best answer gets the brainliest answer.
Answer:
Graph #1
Step-by-step explanation:
In this distance v time graph, a negative slope means moving toward Mr. Wilson, a positive slope means moving away from Mr. Wilson, and a flat section means staying at a particular location without moving.
Write a proof
given: angle 1 and angle 3 are supplementary
prove: m is parallel to n
Answer:
angle 2 is equal to angle 1 ... vertically opposite angles equal
therefore angle 2 and angle 3 are supplementary...co-interior angles supplementary
therefore m is parallel to n
can so.eone explain to me by what bisecting intervals are?
When you bisect something, you cut it into two equally sized pieces. (from Latin: "bi" = two, "sect" = cut)
Bisecting an interval creates two smaller intervals each with half the length of the original interval. Some examples:
• bisecting [0, 2] gives the intervals [0, 1] and [1, 2]
• bisecting [-1, 1] gives the intervals [-1, 0] and [0, 1]
• bisecting an arbitrary interval [tex][a,b][/tex] gives the intervals [tex]\left[a,\frac{a+b}2\right][/tex] and [tex]\left[\frac{a+b}2,b\right][/tex]
Find Lm=
Help me please thanks
A triangle is a three-edged polygon with three vertices. The length of LM is 3√3 cms.
What is a triangle?A triangle is a three-edged polygon with three vertices. It is a fundamental form in geometry. The sum of all the angles of a triangle is always equal to 180°.
The triangle formed by MN, MO, and NO is an equilateral triangle, with the length of each side of the triangle equal to the radius of the triangle. Therefore, the length of MN will be,
MN = r
Now, the length of arc MN can be written as,
2 × π × r × (60° /360°) = π
r = 360 / (60 × 2)
r = 3 cms
Further, the radius of the circle can be written as,
LO = ON = OM = MN = Radius = 3 cms
Since, the triangle formed in a semicircle is a right-angle triangle, therefore, using the Pythagorean theorem we can write,
LM² + MN² = LN²
LM² + MN² = (LO + ON)²
LM² + r² = (r+r)²
LM² + r² = 4r²
LM² = 3r²
LM² = 3(3²)
LM = √27 cms
LM = 3√3 cms
Hence, the length of LM is 3√3 cms.
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Write the following using index notation:
a) 4×4
b) 5x5x5x5
c) 3 × 7 × 7 × 7 × 7
d) 2× 9 × 9 × 9×2×9
Step-by-step explanation:
4^2
5^4
3^1×7^4
2^2 ×9^4
A group of people are having a dinner that costs $160, which is to be shared equally. If two more persons join the group and share the same cost, each will pay $4 less. Find the original number of people in the group.
Answer:
8 people in the original group.
Step-by-step explanation:
Let x be the number of people who were originally present for the dinner that cost $160. Their average cost per person would be:
$160/x
If 2 more people join, there are x+2 people. Ignoring the awkward situation that they apparently did not order any additional food, they agree to share the cost of the dinner. The average cost is now:
$160/(x+2)
We are told that this new average is now $4 less per person.
$160/(x+2) = ($160/x) - $4
$160 = (($160/x) - $4)(x+2)
x = 8 people
Given: AB = DC, AC = DB
Prove: AABC= ADCB
Look at the proof. Name the postulate you would use to prove the two triangles are congruent.
Given: AB DC, AC = DB
Prove: AABC= ADCB
AB= DC
Given
AC DB
Given
BC BC
Reflexive Property
of Congruence
AABC ADCB
?
B
D
C
Answer:
SSS Postulate
Step-by-step explanation:
AB=CD (Given)
AC=BD (Given)
BC=BC (Given)
So SSS Postulate
Side Side Side postulate
Both triangles are congruent by ''SSS Postulates.''
What is mean by Triangle?A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.
Given that;
AB = DC, AC = DB
And, To prove that, ΔABC ≅ ΔDCB
Since, We have;
AB = DC,
AC = DB
And, BC = BC (common sides)
Hence, Both triangles are congruent by ''SSS Postulates.''
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Car A and car B are 60 miles apart. If they start driving towards each other with car A going at twice the speed of car B, how far away will they be from car B's original starting point when they pass each other
The distance between car B's original point and the point where car A and car B pass each other is 20 miles, given the speed of car A is twice the speed of car B, and they are 60 miles apart initially.
We assume the speed of car B to be x miles/hour.
Given car A going at twice the speed of car B, the speed of car A = 2x miles/hour.
When moving toward each other, the relative speed of the cars is the sum of their speeds.
Therefore, the relative speed = x + 2x miles/hour = 3x miles/hour.
The total distance between the cars is given as 60 miles.
Therefore, the time taken by car A and car B to reach the same point = 60/3x hours = 20/x hours.
Therefore, the distance from car B's original point and the point where car A and car B pass each other = Speed of car B*Time = x*(20/x) miles = 20 miles.
Therefore, the distance between car B's original point and the point where car A and car B pass each other is 20 miles, given the speed of car A is twice the speed of car B, and they are 60 miles apart initially.
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Heather, Eric, and Chang sent a total of 116 text messages over their cell phones during the weekend. Chang sent 3 times as many messages as Eric. Eric sent 9 more messages than Heather. How many messages did they each send?
Answer:
Step-by-step explanation:
Declaration
Let Heather's number of messages = x
Let Eric's number of messages = x + 9
Let Chang's number of messages = 3(x +9)
Equation
x + x + 9 + 3(x + 9) = 116
Solution
x + x + 9 + 3(x + 9) = 116 Remove the brackets
x + x + 9 + 3x + 27 = 116 Combine like terms
x + x + 3x + 9 + 27 = 116
5x + 36 = 116 Subtract 36 from both sides
5x + 36 - 36 = 116 - 36 Combine
5x = 80 Divide both sides by 5
5x/5 = 80/5
x = 16
Answer
Heather = 16
Eric = 16 + 9 = 25
Chang = 3*25 = 75
Which equation represents a line that passes through the two points in the
table?
Write a compound inequality that represents the given graph:
OA.x < -2 and x ≥ 4
OB.x < -2 or x ≥4
OC.-2
OD.-2≥x>4
Answer: the correct answer is choice A
Step-by-step explanation:
This is because the inequality symbols are pointed the correct way. The symbols are also showing the correct answers with the greater than or equal to sign is on the correct point value. You find the place for the equal to or greater/less than symbol by the open or closed circle point. An open circle means the number is not included in the answer while a closed circle is included in the answer.
x^3 - y^3 = 218 where x and y are positive integers. Find x+y
The value of x + y is 12
How to solve for x + y?The equation is given as:
x^3 - y^3 = 218
Make x^3 the subject
x^3 = 218 + y^3
The next cubic expression greater than 218 is 343.
So, we assume
x^3 = 343
This gives
x = 7
Substitute x^3 = 343 in x^3 = 218 + y^3
343 = 218 + y^3
Evaluate the like terms
y^3 = 125
This gives
y = 5
So, we have:
x + y = 7 + 5
Evaluate
x + y = 12
Hence, the value of x + y is 12
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Help pls asap
If sine < 0 and tans > 0 then:
OA. 180° << 270°
OB. 0° <9< 90°
C. 270° << 360°
OD. 90° << 180°
Answer:
The answer is C. 270° < ∅ < 360°
Step-by-step explanation:
Given information:
sine < 0 and tans > 0.
Quadrant where sine < 0 is either III or IV.
Quadrant where tans > 0 is either I or III.
It is Quadrant III that has both sines < 0 and tans > 0.
I am not able to solve this question
Answer:
C
Step-by-step explanation:
join my stream for an explanation
(ChildofHermes)
please mark brainliest
im only streaming for 15 more hours
Answer:
Choice C
Step-by-step explanation:
2654^0 and 5^0
Which is greater?
There's a law of exponents for solving this type of question.
The law states that,
x^0 = 1Same case here.
2654^01And,
5^0 1Hence, they are same.
2654⁰ = 5⁰
quadratic formula 2x^2-9x-35=0
Hello,
2x² - 9x - 35 = 0
a = 2 ; b = -9 ; c = -35
∆ = b² - 4ac = (-9)² - 4 × 2 × (-35) = 361 > 0
x1 = (-b - √∆)/2a = (9 - 19)/4 = -10/4 = -5/2
x2 = (-b + √∆)/2a = (9 + 19)/4 = 28/4 = 7
S = {-5/2 ; 7}
Answer:
So x = -5/2, 7.
Step-by-step explanation:
2x^2 - 9x - 35 = 0
Let's see if we can factor this.
We need 2 numbers whose product is 2* -35 ( that is -70) and whose sum is -9.
They are -14 and 5, so we write:
2x^2 - 14x + 5x - 35 = 0
Now we factor by grouping:
2x(x - 7) + 5(x - 7) = 0
(2x + 5)(x - 7) = 0
So x = -5/2, 7.
A company finds that its sales since the company started in 2000 can be modelled by the function s(t)=(20t^2+800t+300)/(8t^2+10t+100) where s is the total sales, in millions of dollars, and t is the number of years since 2000. a) calculate the years when the sales are 9 million, algebraically.
Solving the quadratic function, the sales are of 9 million in the years of 2000 and 2012.
What is a quadratic function?A quadratic function is given according to the following rule:
[tex]y = ax^2 + bx + c[/tex]
The solutions are:
[tex]x_1 = \frac{-b + \sqrt{\Delta}}{2a}[/tex]
[tex]x_2 = \frac{-b - \sqrt{\Delta}}{2a}[/tex]
In which:
[tex]\Delta = b^2 - 4ac[/tex]
The number of sales(in millions of dollars), in t years after 2000, is modeled by the following function:
[tex]S(t) = \frac{20t^2 + 800t + 300}{8t^2 + 10t + 100}[/tex]
The sales are of 9 million when S(t) = 9, hence:
[tex]S(t) = \frac{20t^2 + 800t + 300}{8t^2 + 10t + 100}[/tex]
[tex]9 = \frac{20t^2 + 800t + 300}{8t^2 + 10t + 100}[/tex]
[tex]72t^2 + 90t + 900 = 20t^2 + 800t + 300[/tex]
[tex]52t^2 - 710t + 600 = 0[/tex]
Which is a quadratic equation with coefficients a = 52, b = -710, c = 600, hence:
[tex]\Delta = (-710)^2 - 4(52)(600) = 379300[/tex]
[tex]x_1 = \frac{710 + \sqrt{379300}}{104} = 12.7[/tex]
[tex]x_2 = \frac{710 - \sqrt{379300}}{104} = 0.91[/tex]
t is measured in years after 2000, the sales are of 9 million in the years of 2000 and 2012.
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Use the law of cosines to find the value of cos0. Round your answer to two decimal places.
A.0.22
B.0.84
C.0.35
D.-1.04
Using the law of cosines on the given triangle, we see that; cosθ = 0.22
How to use the law of cosines?The law of cosines is expressed as;
c² = a² + b² - 2ab cos C
Now, in this case the angle is θ and employing this law of cosines gives us;
10.2² = 5.7² + 9.8² - 2(5.7 * 9.8)*cosθ
104.04 = 32.49 + 96.04 - 111.72 * cosθ
104.04 = 128.53 - 111.72 * cosθ
(128.53 - 104.04)/111.72 = cosθ
cosθ = 0.22
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TRY IT Identifying Matrix Attributes
What is the size of matrix A? ✓ 3 x 5
COMPLETE
How many elements are in the matrix? ✔ 15
COMPLETE
How many rows are in the matrix? 3
COMPLETE
Where is the largest element in the matrix located?
Oa12
x a34
a21
0 12 7 22 197
A = 540 090
33 5 0 52 0
The location of the largest element in the matrix is a₂₁, which is 54. The location of the largest element in the matrix is a₂₁, which is 54.
What is the matrix?Matrix is defined as the group of numerical data, functions, and complex numbers in a specific way such that the representation array looks like a square, rectangle shape.
The following details about the matrix can be written,
The size of matrix A is 3x5.The number of elements in the given matrix is 15.There are 3 rows in the matrix.The location of the largest element in the matrix is a₂₁, which is 54.
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Eliza drew scale drawing of her bedroom. The drawing shows a room that is 4“ x 3“. The scale she used is 1 inch: 4 inch. What is the area of her actual bedroom?
Area of Eliza's bedroom = 192 sq. inch
What is Scaling ?Scaling is a method used in measurement when large data has to be shown in a small area.
It is given that
The drawing shows a room that is 4“ x 3“
The scale used is 1 inch: 4 inch
So the actual dimensions will be 16'' x 12"
The area of a rectangle is Length * Breadth
Area of her bedroom = 16 * 12
Area of her bedroom = 192 sq. inch
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What is the missing step in solving the inequality 4(x – 3) + 4 < 10 + 6x?
1. The distributive property: 4x – 12 + 4 < 10 + 6x
2. Combine like terms: 4x – 8 < 10 + 6x
3. The addition property of inequality: 4x < 18 + 6x
4. The subtraction property of inequality: –2x < 18
5. The division property of inequality: ________
The missing step in solving the inequality is x > -9
Solving inequalitiesFrom the question, we are to determine the missing step in solving the given inequality
1. The distributive property: 4x – 12 + 4 < 10 + 6x
2. Combine like terms: 4x – 8 < 10 + 6x
3. The addition property of inequality: 4x < 18 + 6x
4. The subtraction property of inequality: –2x < 18
5. The division property of inequality: ________
Applying the division property of inequality
-2x/-2 < 18/-2
x > -9
Hence, the missing step in solving the inequality is x > -9
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Everyday Math
583. Coin production In 1942 , the U.S. Mint produced 154,500,000 nickels. Write 154,500,000$in scientific notation.
Answer:
In 1942 , the U.S. Mint produced 154,500,000 nickels. The scientific notation of 154,500,000 is obtained as [tex]$1.545 \times 10^{8}$[/tex].
Step-by-step explanation:
In the problem, it is given that in 1942, the U.S. Mint produced 154,500,000 nickels.It is required to write 154,500,000 in scientific notation.To solve this first, move the decimal point so that the first factor is greater than or equal to 1 also it should be less than 10.Count the number of decimal places n, and write the number as a product with a power of 10. Then check the answer.Step 1 of 2
Obtain the scientific notation.
Here the original number, 154,500,000 is greater than 1 so we will have a positive power of 10 .
Move the decimal point to get 1.545, a number between 1 and 10 . Then we get,
1.54500000
The count of the number of decimal places the point was moved is 8 .
Write as a product with a positive power of 10 .
[tex]$1.545 \times 10^{8}$[/tex]
Step 2 of 2
Check the answer.
Rewrite and simplify to check our answer.
[tex]$$\begin{gathered}1.545 \times 10^{8}=1.545 \times 100000000 \\=154500000\end{gathered}$$[/tex]
Two lengths of a triangle are shown.
Triangle A B C. Side A B has a length of 6 feet and side A C has a length of 6 feet.
Which could be the length of Side B C?
The answer choice which could be the length of side BC is; BC < 12.
Which answer choice could be the length of side BC?It follows from the task content that the possible length of the side BC is to be determined.
Since, the sides given AB and BC both have length sum equal to 12. It follows from the triangle inequalities that the length of the third side must be less than the sum of lengths of the other two sides.
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A quantity with an initial value of 130 decays exponentially at a rate such that the
quantity cuts in half every 9 weeks. What is the value of the quantity after 21 days
A quantity with an initial value of 130 decays exponentially at a rate such that the quantity cuts in half every 9 weeks. 103.18 is the value of the quantity after 21 days.
What is exponential decay?A process known as exponential decay is when a quantity diminishes over time at a pace that reduces proportionately as the amount shrinks.
The term "exponential decay" refers to a population's or group's rate of decline when it is proportionate to the size of the population. The overall value falls in exponential decay, yet the percentage that departs stays constant over time.
Let, y = 130. aˣ
here, x = number of days
Given that,
A quantity with an initial value of 130 decays exponentially.
the quantity cuts in half every 9 weeks
the value of the quantity after 21 days = ?
So, 130 × [tex]a^{(7\times9) }[/tex] = 130 × (1/2)
or, a⁶³ = 1/2
or, a [tex](1/2)^{(1/63)}[/tex]
Thus, y = 130 × [tex](1/2)^{(x/63)}[/tex]
when x = 21
thus, y = 130 ×[tex](1/2)^{(21/63)}[/tex]
y = 130 × [tex](1/2)^{(1/3)}[/tex]
y ≈ 103.18
Hence, 103.18 is the value of the quantity after 21 days.
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Dana is buying Grandparents' Day gifts at the mall. She spends $13.75 on a teacup for her grandmother. She spends 4/5 of that on a coffee mug for her grandfather. How much money does Dana spend in all?
The total money Dana spent in all is $24.75
Ratio and proportionsRatio are written as fractions of two integers. For instance a/b.
Amount spent for grandmother = $13.75
Amount spent on grand father = 4/5 * 13.75
Amount spent on grand father = $11
Take the sum
Total money spent = $13.75 + $11
Total money spent = $24.75
Hence the total money Dana spent in all is $24.75
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