Step-by-step explanation:
a = x^2 - 1
b = 2x
c = x^2 + 1
The Pythagorean theorem states
a^2 + b^2 = c^2
Let's find a^2 and b^2 and add them to get a^2 + b^2:
a^2 = (x^2 - 1)^2 = x^4 - 2x^2 + 1
b^2 = (2x)^2 = 4x^2
a^2 + b^2 = x^4 - 2x^2 + 1 + 4x^2 = x^4 + 2x^2 + 1
Now let's find c^2:
c = x^2 + 1
c^2 = (x^2 + 1)^2 = x^4 + 2x^2 + 1
We see that both a^2 + b^2 and c^2 equal x^4 + 2x^2 + 1, so we have shown that the triangle is a right triangle.
We can see that a² + b² = c² = x⁴ + 2x² + 1, indicating that the triangle is a right triangle.
What is the definition of a right-angle triangle?
It's a form of a triangle with one 90-degree angle that follows Pythagoras' theorem and can be solved using the trigonometric function.
According to Pythagoras, the sum of the squares of two sides equals the square of the longest side.
Given data;
a = x² - 1
b = 2x
c = x² + 1
According to the Pythagorean theorem;
[tex]\rm a^2 + b^2 = c^2[/tex]
Let's find a^2 and b^2 and add them to get a^2 + b^2:
[tex]\rm a^2 = (x^2 - 1)^2 \\\\ a^2 = x^4 - 2x^2 + 1 \\\\ b^2 = (2x)^2 \\\\ b^2 = 4x^2\\\\[/tex]
Left hand side:
[tex]\rm a^2 + b^2 = x^4 - 2x^2 + 1 + 4x^2 \\\\ a^2 + b^2 =x^4 + 2x^2 + 1[/tex]
Right hand side:
[tex]\rm c^2 = (x^2 + 1)^2 \\\\ c^2 = x^4 + 2x^2 + 1[/tex]
L.HS.=R.H.S
Hence,the given triangle is a right angled triangle.
To learn more about right-angle triangles, refer to https://brainly.com/question/3770177.
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hello! i’m currently working on some practice questions, and i was wondering how you would solve this expression!
Answer:
7
Step-by-step explanation:
Givens
x = 4
y = 2
z =-3
equation
xy - z^y
Solution
(4)(2) - (-3)^2 be sure and add the brackets. Otherwise it won't come out correctly.
16 - (9) = 7
cos^2x+cos^2(120°+x)+cos^2(120°-x)
i need this asap. pls help me
Answer:
[tex]\frac{3}{2}[/tex]
Step-by-step explanation:
Using the addition formulae for cosine
cos(x ± y) = cosxcosy ∓ sinxsiny
---------------------------------------------------------------
cos(120 + x) = cos120cosx - sin120sinx
= - cos60cosx - sin60sinx
= - [tex]\frac{1}{2}[/tex] cosx - [tex]\frac{\sqrt{3} }{2}[/tex] sinx
squaring to obtain cos² (120 + x)
= [tex]\frac{1}{4}[/tex]cos²x + [tex]\frac{\sqrt{3} }{2}[/tex]sinxcosx + [tex]\frac{3}{4}[/tex]sin²x
--------------------------------------------------------------------
cos(120 - x) = cos120cosx + sin120sinx
= -cos60cosx + sin60sinx
= - [tex]\frac{1}{2}[/tex]cosx + [tex]\frac{\sqrt{3} }{2}[/tex]sinx
squaring to obtain cos²(120 - x)
= [tex]\frac{1}{4}[/tex]cos²x - [tex]\frac{\sqrt{3} }{2}[/tex]sinxcosx + [tex]\frac{3}{4}[/tex]sin²x
--------------------------------------------------------------------------
Putting it all together
cos²x + [tex]\frac{1}{4}[/tex]cos²x + [tex]\frac{\sqrt{3} }{2}[/tex]sinxcosx + [tex]\frac{3}{4}[/tex]sin²x + [tex]\frac{1}{4}[/tex]cos²x - [tex]\frac{\sqrt{3} }{2}[/tex]sinxcosx + [tex]\frac{3}{4}[/tex]sin²x
= cos²x + [tex]\frac{1}{2}[/tex]cos²x + [tex]\frac{3}{2}[/tex]sin²x
= [tex]\frac{3}{2}[/tex]cos²x + [tex]\frac{3}{2}[/tex]sin²x
= [tex]\frac{3}{2}[/tex](cos²x + sin²x) = [tex]\frac{3}{2}[/tex]
On which number line are -3 and its opposite shown?
Answer:
The number line should look like this;
<---- -3 -------- 0 ------- 3 ----->
It must have both 3 and -3 shown
Step-by-step explanation:
Step-by-step explanation:how do you feguire this out because the left of a number line is always negative and the right is always posittive easy right i hope it helped you just like when i learned it it helped me.
Which congruence theorems prove tht
MNP and QRS are be congruent?
Word Bank:
AAS theorem, ASA postulate, SAS postulate,
SSS postulate, HA, AA, HL, LA, LL, HH
Answer
AAS
Step-by-step explanation:
I need help finding the answer
Answer:
32
Step-by-step explanation:
The formula is base x hight.
The base is 8.
The hight is 4.
You just multiply the two.
8x4=32
Simple. :)
Hope this helps!
Answer: 42 inches squared
Step-by-step explanation:
To find the total area you will find the area of the rectangle and then add that to the area of the triangle. (Remember a triangle is half of a rectangle!) First we will find the area of the rectangle...
4x8=32 inches squares
Now let’s find the area of the triangle, find the rectangle then divide it in half...
5x4=20 inches squared
20/2=10 inches squared
Now add 32 to 10 and you’ll have your answer!
32+10=42 inches squared
what is the solution set of 7x^2 + 3x = 0
Answer:
the answer is x=0,-(3/7)
Step-by-step explanation:
Answer:
x = { - [tex]\frac{3}{7}[/tex], 0 }
Step-by-step explanation:
Given
7x² + 3x = 0 ← factor out x from each term
x(7x + 3) = 0
Equate each factor to zero and solve for x
x = 0
7x + 3 = 0 ⇒ 7x = - 3 ⇒ x = - [tex]\frac{3}{7}[/tex]
66=1/2 h(5+6)
Solve for h
Answer:
h= 12
Step-by-step explanation:
Given
66 = [tex]\frac{1}{2}[/tex] h(5 + 6)
Multiply both sides by 2 to eliminate the fraction
132 = h(5 + 6)
132 = 11h ( divide both sides by 11 )
12 = h
Classify the following triangle check all that apply
Answer: B and F
Hope it helps!!!
Answer: OPTION B AND OPTION D.
Step-by-step explanation:
Analyze the triangle provided.
You can observe that the lenghts of its sides are: 10.9, 15, 14
Then the lenghts of its sides are not equal.
By definition, when all sides of a triangle are different this is called: "Scalene".
You can notice that the measures of its angles are: 63°, 44° and 73°
Then, all its angles are less than 90 degrees.
By definition if all three angles of a triangle are less than 90 degrees, then is called: "Acute".
Three families went to the movies together. The Smiths ordered two tubs of popcorn, one plate of nachos, and three drinks. They spent $65. The Langes ordered three tubs of popcorn, two plates of nachos, and five drinks. They spent $85. The Radfords ordered one tub of popcorn, one plate of nachos, and two drinks. They spent $40. Which system of equations matches their night at the movies?
Answer:
2t + n + 3d = 65
3t + 2n + 5d = 85
t + n + td = 40
Step-by-step explanation:
tubs of popcorn = t
plates of nachos = n
drinks = d
The Smiths: 2t + n + 3d = 65
The Langes: 3t + 2n + 5d = 85
The Radfords: t + n + td = 40
So, the correct answer would be the one including all 3 of these equations. :)
Answer:
The system of equations are:
[tex]2t + n + 3d = 65[/tex]
[tex]3t + 2n + 5d = 85[/tex]
[tex]t + n + 2d = 40[/tex]
Step-by-step explanation:
Consider the provided information.
let "t" represents tubes of popcorn, "n" represents nachos and "d" represents drink.
The Smiths ordered two tubs of popcorn, one plate of nachos, and three drinks. They spent $65. Which can be represents as:
[tex]2t + n + 3d = 65[/tex]
Langes ordered three tubs of popcorn, two plates of nachos, and five drinks. They spent $85. Which can be represents as:
[tex]3t + 2n + 5d = 85[/tex]
The Radfords ordered one tub of popcorn, one plate of nachos, and two drinks. They spent $40. Which can be represents as:
[tex]t + n + 2d = 40[/tex]
Thus, the system of equations are:
[tex]2t + n + 3d = 65[/tex]
[tex]3t + 2n + 5d = 85[/tex]
[tex]t + n + 2d = 40[/tex]
what is the eighth term in the sequence x+3, - 2x^2 - 6x, 4x^3 +12x^2
Answer:
[tex]a_8=-128x^8-384x^7[/tex]
Step-by-step explanation:
The terms of the sequence are:
[tex]x+3,-2x^2-6x,4x^3+12x^2,...[/tex]
We can rewrite the terms in factored form to get;
[tex]x+3,-2x(x+3),4x^2(x+3),...[/tex]
We can see that the subsequent terms are obtained by multiplying the previous term by [tex]-2x[/tex]. This is called the common ratio.
Therefore the first term of this geometric sequence is [tex]a_1=x+3[/tex] and the common ratio is [tex]r=-2x[/tex].
The nth term of a geometric sequence is given by: [tex]a_n=a_1(r^{n-1})[/tex].
Let us substitute the first term, the common ratio, and [tex]n=8[/tex] to obtain:
[tex]a_8=(x+3)(-2x)^{8-1}[/tex]
[tex]a_8=(x+3)(-2x)^{7}[/tex]
[tex]a_8=-128x^7(x+3)[/tex]
[tex]a_8=-128x^8-384x^7[/tex]
Judy works at a candy factory that makes Sugar Rush candy bars. She is in charge of quality control and has to make sure each candy bar has the correct mass. Each candy bar is required to weigh 12 grams, with a tolerance of 0.45 grams. What is the acceptable weight range for each candy bar?
1. Define a variable for this situation.
2. Write the absolute value inequality that describes the acceptable weight range for each candy bar.
3.Solve the absolute value inequality to find the acceptable weight range for each candy bar using numbers and symbols.
4. Using words, describe the possible acceptable weight range for each candy bar.
Step-by-step explanation:
1. x = acceptable weight of candy bar
2. |x - 12| ≤ 0.45
3. x - 12 ≤ 0.45, x - 12 ≥ -0.45
x ≤ 12.45, x ≥ 11.55
4. The acceptable weight range for each candy bar is between 11.55 grams and 12.45 grams.
What is the greatest common factor of 22a2 and 32a
Step-by-step explanation:
Write the prime factorization for each:
22a² = 2×11×a²
32a = 2⁵×a
So the greatest common factor is 2a.
The area of a compact disc is 452 4/7 square centimeters. What is the diameter of a compact disc ? Use 22/7 as an approximation for pie?
[tex]\bf \textit{area of a circle}\\\\ A=\pi r^2~~ \begin{cases} r=radius\\ \cline{1-1} A=452\frac{4}{7} \end{cases}\implies 452\frac{4}{7}=\pi r^2\implies \cfrac{3168}{7}=\pi r^2 \implies \cfrac{3168}{7\pi }=r^2 \\\\\\ \stackrel{\pi =\frac{22}{7}}{\cfrac{3168}{7\cdot \frac{22}{7}}}=r^2\implies \cfrac{3168}{22}=r^2\implies 144=r^2\implies \sqrt{144}=r\implies 12=r \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \stackrel{diameter=2r}{d=24}~\hfill[/tex]
6x - 26 = 58 + 4x, 15 points to whoever solves this
Answer:
x=42
Step-by-step explanation:
Add by 26 from both sides of equation.
6x-26+26=58+4x+26
Simplify.
6x=4x+84
Subtract by 4x from both sides of equation.
6x-4x=4x+84-4x
Simplify.
2x=84
Divide by 2 from both sides of equation.
2x/2=84/2
Simplify, to find the answer.
84/2=42
x=42 is the correct answer.
I hope this helps you, and have a wonderful day!
If there are 6 serving in a 2/3 pound lb package of pound is in each serving
Answer:
[tex]\boxed{\math{\frac{1}{9}\text{ lb}}}[/tex]
Step-by-step explanation:
[tex]\text{Size of one serving} = \dfrac{\text{Total size}}{\text{No of servings}} = \dfrac{\frac{2}{3}\text{ lb}}{\text{6 servings}}\\\\\text{Change the 6 to a fraction and change divide to multiply}\\\\\dfrac{2}{3} \div 6 = \dfrac{2}{3} \times \dfrac{1}{6}[/tex]
[tex]\text{Cancel the 2s}\\\\\dfrac{2}{3} \times \dfrac{1}{6} = \dfrac{1}{3} \times \dfrac{1}{3}\\\\\text{Multiply numerators and denominators}\\\\\dfrac{1}{3} \times \dfrac{1}{3} = \dfrac{1}{9}\\\\\text{Each serving contains }\boxed{\mathbf{\frac{1}{9}\textbf{ lb}}}[/tex]
What is the solution of the graph?
Answer:
no solution
Step-by-step explanation:
The solution to a system of equations given graphically is at the point of intersection of the 2 graphs.
The given lines are parallel ( both have a slope = 2 )
Hence the lines never intersect thus there is no solution.
A sequence is defined by the recursive function
f(n + 1) = f(n) - 2
if f(1) = 10, what is f(3)?
1
6
8
30
Answer:
6Step-by-step explanation:
[tex]f(n+1)=f(n)-2\\\\f(1)=10\\f(2)=f(1)-2=10-2=8\\f(3)=f(2)-2=8-2=6[/tex]
Answer:
B) 6
Step-by-step explanation:
Explain how you can use equivalent fractions to find the quotient of 2 3 ÷ 4.
Answer:
see below
Step-by-step explanation:
2/3 ÷ 4
We use copy dot flip
The flip means make a reciprocal of the second number
2/3 * 1/4
Multiply the numerators
2*1 = 2
Multiply the denominators
3*4 =12
Put the numerator over the denominator
2/12
Simplify
1/6
To use equivalent fractions to find the quotient of \( \frac{2}{3} \) ÷ 4, follow these steps:
### Step 1: Understand the Operation
Division of fractions can be thought of as multiplying by the reciprocal. The reciprocal of a number a is simply \( \frac{1}{a} \).
### Step 2: Find the Reciprocal of the Divisor
The divisor here is the whole number 4. Its reciprocal is \( \frac{1}{4} \).
### Step 3: Multiply the Dividend by the Reciprocal of the Divisor
Instead of dividing \( \frac{2}{3} \) by 4, you can multiply \( \frac{2}{3} \) by \( \frac{1}{4} \).
### Step 4: Perform the Multiplication
Now, multiply the two fractions:
\[ \frac{2}{3} \times \frac{1}{4} = \frac{2 \cdot 1}{3 \cdot 4} \]
This results in a new fraction:
\[ \frac{2}{12} \]
### Step 5: Simplify the Resulting Fraction
Finally, you need to simplify the fraction to its simplest form. To do that, find the greatest common factor (GCF) of the numerator and the denominator and divide both by this number.
For \( \frac{2}{12} \), the greatest common factor is 2. So we divide both the numerator and the denominator by 2:
\[ \frac{2 \div 2}{12 \div 2} = \frac{1}{6} \]
### Conclusion
Therefore, the quotient of \( \frac{2}{3} \) ÷ 4 is \( \frac{1}{6} \). This is the simplest form of the fraction you obtain when \( \frac{2}{3} \) is divided by 4.
Your phone plan charges you an initial fee and then a certain amount depending on the amount of data you use they send you periodic updates of your insiste and your current bill cost. After using 3 GB of data you owe $30. After 5 GB of data you owe $40. What is the initial fee prior to the data usage charge ?
Answer:
5
Step-by-step explanation:
dot
Simplify the expression.
(7.46)** . (7.46)
Answer:
D. 7.46
Hope this helps and have a nice day!!
If I'm wrong pleaseeee tell me
Step-by-step explanation:
Which is correct? I am marking Brainliest.
Answer:
A. concave hexagon
Step-by-step explanation:
It has 6 sides, so it's a hexagon. (A heptagon has 7 sides.)
If all interior angles are less than 180 deg, then it is convex. There is one interior angle on the left side that is more than 180 deg, so it is concave.
Answer: concave hexagon
Consider the paragraph proof.
Given: D is the midpoint of AB, and E is the midpoint of AC.
Prove:DE = BC
It is given that D is the midpoint of AB and E is the midpoint of AC. To prove that DE is half the length of BC, the distance formula, d = , can be used to determine the lengths of the two segments. The length of BC can be determined with the equation BC = , which simplifies to 2a. The length of DE can be determined with the equation DE = , which simplifies to ________. Therefore, BC is twice DE, and DE is half BC.
Which is the missing information in the proof?
a
4a
a2
4a2
Answer:
a
Step-by-step explanation:
You're trying to find the distance between D and E so u use the distance formula.
sqrt (a+b-b^2)+(c-c)^2=sqrt a^2=a
Answer:
a
Step-by-step explanation:
We are given that
D is the mid-point of AB and E is the mid-point of AC.
We have to find the missing information in given proof of DE is equal to half of BC.
Proof:
D is the mid-point of AB and E is the mid-point of AC.
The coordinates of A are (2b,2c)
The coordinates of D are (b,c)
The coordinates of E are (a+b,c)
The coordinates of B are (0,0)
The coordinates of C are (2a,0)
Distance formula:[tex]\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Using the formula
Length of BC=[tex]\sqrt{(2a)^2+(0-0)^2}=2a[/tex] units
Length of DE=[tex]\sqrt{(a+b-b)^2+(c-c)^2}=a[/tex] units
[tex]BC=2a=2\times DE[/tex]
[tex]DE=\frac{1}{2}BC[/tex]
Hence, proved.
Option A is true.
Given ƒ(x) = 7x + 1 and g(x) = x^2, find (g ○ ƒ)(x)
For this case we have the following functions:
[tex]f (x) = 7x + 1\\g (x) = x ^ 2[/tex]
We must find[tex](g_ {0} f) (x).[/tex]
By definition of composition of functions we have to:
[tex](g_ {0} f) (x) = g (f (x))[/tex]
So:
[tex](g_ {0} f) (x) = g (f (x)) = (7x + 1) ^ 2 = 49x ^ 2 + 2 (7x) (1) + 1 = 49x ^ 2 + 14x + 1[/tex]
ANswer:
[tex](7x + 1) ^ 2 = 49x ^ 2 + 14x + 1[/tex]
algebra II engenuity
Answer:
First Option
Step-by-step explanation:
Given expression is:
[tex]\sqrt[4]{x^{10}}[/tex]
The radicand's exponent will be made multiple of 4 to make the calculations easy
So,
[tex]= \sqrt[4]{x^8 * x^2}[/tex]
The 4 outside the radical means that the power that will be multiplied with the exponents will be 1/4
So,
[tex]= x^{(8*\frac{1}{4})} * x^{(2*\frac{1}{4})}\\=x^2 \sqrt[4]{x^2}[/tex]
As x^2 couldn't be solved using radical, it will remain inside the radical.
So the correct answer is first option..
Answer: First option.
Step-by-step explanation:
Knwing that we must find which is the equivalent expression of the expression [tex]\sqrt[4]{x^{10}}[/tex], it is important to remember the Product of powers property, which states the following:
[tex](a^m)(a^n)=a^{(m+n)}[/tex]
The we can rewrite the expression:
[tex]=\sqrt[4]{x^8x^2}[/tex]
Remember that:
[tex]\sqrt[n]{a^n}=a[/tex]
Then we get this equivalent expression:
[tex]=x^2(\sqrt[4]{x^2})[/tex]
y+1=-3x(x+4)
help me solve please
Which expression will simplify to 1?
Let us check each Option :
[tex]\mathsf{First\;Option : \left(\dfrac{m + 9}{m - 9}\right)\left(\dfrac{m + 9}{m - 9}\right)}[/tex]
[tex]\mathsf{\implies \left(\dfrac{m + 9}{m - 9}\right)^2\;\neq\;1}[/tex]
[tex]\mathsf{Second\;Option : \left(\dfrac{m + 9}{m - 9}\right)\left(\dfrac{m - 9}{m + 9}\right)}[/tex]
[tex]\mathsf{\implies 1}[/tex]
[tex]\mathsf{Third\;Option : \left(\dfrac{m + 9}{m - 9}\right)\left(\dfrac{9 + m}{9 - m}\right)}[/tex]
[tex]\mathsf{\implies \left(\dfrac{m + 9}{m - 9}\right)\left(\dfrac{m + 9}{-(m - 9)}\right)}[/tex]
[tex]\mathsf{\implies-\left(\dfrac{m + 9}{m - 9}\right)\left(\dfrac{m + 9}{m - 9}\right)}[/tex]
[tex]\mathsf{\implies-\left(\dfrac{m + 9}{m - 9}\right)^2\;\neq\;1}[/tex]
[tex]\mathsf{Fourth\;Option : \left(\dfrac{m + 9}{m - 9}\right)\left(\dfrac{9 - m}{9 + m}\right)}[/tex]
[tex]\mathsf{\implies \left(\dfrac{m + 9}{m - 9}\right)\left(\dfrac{-(m - 9)}{9 + m}\right)}[/tex]
[tex]\mathsf{\implies -\left(\dfrac{m + 9}{m - 9}\right)\left(\dfrac{m - 9}{9 + m}\right)}[/tex]
[tex]\mathsf{\implies -1\;\neq\;1}[/tex]
Answer : Option (2)
Answer:
B.⁽[tex](\frac{m}{m}\frac{+9}{-9} ) (\frac{m}{m} \frac{- 9}{+ 9} )[/tex]
Step-by-step explanation:
Divide simplify your answer
Answer:
[tex]\large\boxed{\dfrac{2s-6}{s+3}}[/tex]
Step-by-step explanation:
[tex]s^2-9=s^2-3^2\qquad\text{use}\ a^2-b^2=(a-b)(a+b)\\=(s-3)(s+3)\\\\s^2+6s+9=s^2+2(s)(3)+3^2\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\=(s+3)^2[/tex]
[tex]\dfrac{s^2-9}{2s}\div\dfrac{s^2+6s+9}{4s}=\dfrac{s^2-9}{2s\!\!\!\!\diagup_1}\cdot\dfrac{4s}{s^2+6s+9}\\\\=\dfrac{(s-3)(s+3)}{2s\!\!\!\!\!\diagup_{_1}}\cdot\dfrac{4s\!\!\!\!\!\diagup^{^2}}{(s+3)^2}=\dfrac{2(s-3)(s+3)}{(s+3)(s+3)}\qquad\text{cancel}\ (s+3)\\\\=\dfrac{2(s-3)}{s+3}=\dfrac{2s-6}{s+3}[/tex]
A vertex a is at (-1, -2)
Answer:
A'(-4,4)
B'(-2,11)
Step-by-step explanation:
6 unit up and 3 unit left
A (-1,-2) : 6 unit up → (-1,4) → 3 unit left → (-4,4)
A'(-4,4)
B (1,5) : 6 unit up → (1,11) → 3 unit left → (-2,11)
B'(-2,11)
Can someone help me plz and the last two that you couldn’t see was ( c- 1 1/3 ) and ( D- 1 1/9)
Answer:
B 9/10
Step-by-step explanation:
3/5 ÷2/3
Copy dot flip
3/5 * 3/2
9/10
The graph of relation r is shown. Which of the following graphs represents the relation and it’s inverse
Answer:
To easily solve this question, we must realize that the graph of the relation is very similar to that of the expression
y = √(x-a) , where a>0
If we take a look at the image attached, we have plotted the graph of
y = √(x-1) , and its correspondent inverse function.
This means that the answer is the first option