Answer:
The answer above is completely correct, but it may help the student to see how to reach it:
We seek a square root of
3
−
2
√
2
. This must have the form
a
+
b
√
2
.
(
a
+
b
√
2
)
2
=
a
2
+
2
a
b
√
2
+
2
b
2
So
a
2
+
2
b
2
=
3
and
2
a
b
=
−
2
⇒
a
b
=
−
1
These are two simultaneous equations for
a
and
b
.
Rearrange the second:
b
=
−
1
a
Substitute into the first:
a
2
+
2
a
2
=
3
a
4
+
2
=
3
a
2
a
4
−
3
a
2
+
2
=
0
Note that this is a quadratic in
a
2
:
(
a
2
)
2
−
3
(
a
2
)
+
2
=
0
Factorise:
(
a
2
−
2
)
(
a
2
−
1
)
=
0
This gives us two possible solutions for
a
2
:
2
and
1
, and so the four solutions for
a
:
±
√
2
and
±
1
.
We are looking for integer solutions for
a
, and so
±
1
are possible solutions. But the other two are possible too - they can simply be folded in to the
√
2
term. This wouldn't have been possible if we'd had the root of some other number in the solution for
a
, but this solution is a special case.
Now use the second equation to deduce the four equivalent solutions for
b
:
b
=
−
1
a
b
=
¯¯¯¯¯
+
1
√
2
=
¯¯¯¯¯
+
1
2
√
2
and
¯¯¯¯
+
1
.
So we have the four solution pairs
(
a
,
b
)
:
(
√
2
,
−
1
2
√
2
)
(
−
√
2
,
1
2
√
2
)
(
1
,
−
1
)
(
−
1
,
1
)
This is a bit suspicious - we expect only two solutions, positive and negative square roots, so we wonder if some of these are identical to each other: When we substitute them in to our desired expression
a
+
b
√
2
, we get:
√
2
−
1
2
√
2
√
2
=
−
1
+
√
2
−
√
2
+
1
2
√
2
√
2
=
1
−
√
2
1
−
√
2
−
1
+
√
2
So the two solutions with
√
2
are identical to the two simpler solutions, so we can get rid of them. We now have two solutions, positive and negative square roots:
1
−
√
2
−
1
+
√
2
When we take the written square root of a quantity, it is implied that the desired root is the positive root, the "principal value" of the square root function. So we take the single solution that comes from
a
=
+
1
:
1
−
√
2
Double check: Make sure that this produces the desired answer:
(
1
−
√
2
)
2
=
1
−
2
√
2
+
2
=
3
−
2
√
2
I think this will help you
The volume of this cube is 64 cubic feet. What is the value of a?
a rectangular rug is 1.8m long 0.7 wide, all measurement are given to 1 decimal place
(They have been tranucated and not rounded)
A) state the minimum possible length of the rug
B) calculate the maximum possible area of the rug
Correct answer gets 5 star
Answer:
A. The function is Nonlinear.
A group of 13 friends all want to ride a roller coaster. The coaster seats 15 in a single file row from front to back. How many ways can they arrange themselves?
Answer:
Step-by-step explanation:
Evaluate the expression when a=-7 and x=6 .
X-2a
Answer:
=X-2a
= 6- 2× (-7)
= 6 + 14
= 20
✌✌✌✌
I need help, can someone help me. please
Answer:
y = - 1/3x
Step-by-step explanation:
So the equation of a line can be written in the form y= mx+ c
Here y is the value of y in the y axis, x is the value of x in the x axis, m is the gradient ad c is the co-efficient.
We need to find the gradient of the line which is - 1/3.
That makes our equation y = - 1/3x
HOPE THIS HELPED
a polynomial has been factored below but some constants are missing. 2x^3-8x^2-24x=ax(x+b)(x+c)
Answer:
The polynomial is 2x^3 - 8x^2 - 24x
And we can factor out a 2x from each of the three terms:
2x(x^2 - 4x - 12)
Lastly, factor the remaining quadratic:
2x(x+(-2))(x+6)
And we have our answer:
a=2
b=-2
c=6
Let me know if this helps!
Answer:
a =2, b =2, and c = -6
Step-by-step explanation:
We factor the polynomial and then see which value corresponds to what.
2x^3-8x^2-24x
As we see it, all terms are factorable by 2x. So if we take out 2x from every term, we get
2x(x^2 - 4x - 12)
Now we factor the quadratic, which we can do mentally to get
2x(x+2)(x-6)
ax(x+b)(x+c)
Comparing that to ax(x+b)(x+c), we can tell that a =2, b =2, and c = -6.
|Large{\frac {x}{x + 3} = \frac {5}{2}}
Answer:
x=-5
Step-by-step explanation:
[tex]\frac{x}{x+3} =\frac{5}{2} \\5x+15=2x\\5x-2x=-15\\3x=-15\\x=-15/3=-5[/tex]
Julie wants to make 800 g of a 15% alcohol solution by mixing a
20% solution and a 4% solution. How many grams of each kind
does she need?
Answer:
x =550
y=250
Step-by-step explanation:
.2x + .04y = 120
x + y = 800
What is the distance between (1, 9) and (1, -4)? Use the distance formula.
D=(x2-x1)2+(y2-y1)2
Answer:
D. 13
Step-by-step explanation:
You just have to plug in numbers then solve.
(1,9) and (1,-4). Every coordinate is placed as (x,y)
(1,9) is labeled as (x1, y1)
(1,-4) is labeled as (x2,y2)
Now plug in the numbers.
d= Square root of (1-1)^2 + (-4-9)^2
d= Square root of 1^2 +-13^2
d= Square root of 1+169
d= Square root of 170
d= 13.04
Rounded to a whole number, the answer is 13
Answer:
D
Step-by-step explanation:
you have to substitute the values in the distance formula,in this case x2 is 1,x1 is 1,y2 is -4 and y1 is 9
d=√(x2-x1)²+(y2-y1)²
=√(1-1)²+(-4-9)²
=√(0)²+(-13)²
=√169
=13
I hope this helps
(I'm so confused please help)
Suppose M is the midpoint of XY. If XM = 2x + 11 and MY = 5x - 16, find the value of X.
A) x = -4
B) x = 4
C) x = -9
D) x = 9
E) x = 2
My math:
2x + 11 = 5x + 16
(-5x and -11 from both sides)
-3x=5
(divide -3 on each side)
x=-5/3
But it isn't an option, so I don't know.
Answer:
D
Step-by-step explanation:
Instead of 2x + 11 = 5x + 16, you should have put 2x+11 = 5x - 16. There is a sign change right before the 16, and you may have accidentally written down +16 instead of -16. I think that's where the problem is.
2x + 11 = 5x - 16
subtract 5x from both sides to make all variables on one side
-3x + 11 = -16
subtract 11 from both sides to isolate the x and its coefficient
-3x = -27
divide both sides by -3 to isolate x
x = 9
4. Maddie is making pineapple punch for her birthday party. The recipe says you need 3 cups of pineapple
juice for every 4 cups of ginger ale. If she uses 16 cups of ginger ale, how much pineapple juice will she
need?
simplify: 3(x-2)+2(2x+5)=35
Answer:
x=31/7
Step-by-step explanation:
3x-6+4x+10=35
7x=31
x=31/7 (The task was to find the x, not simplify)
Answer:
[tex]x=\frac{31}{7}[/tex]
Step-by-step explanation:
One is given the following equation,
[tex]3(x-2)+2(2x+5)=35[/tex]
Distribute, multiply every term inside the parenthesis by the term outside of it.
[tex]3(x-2)+2(2x+5)=35[/tex]
[tex](3)(x)-(3)(2)+(2)(2x)+(2)(5)=35[/tex]
Simplify,
[tex](3)(x)-(3)(2)+(2)(2x)+(2)(5)=35[/tex]
[tex]3x-6+4x+10=35[/tex]
[tex]7x+4=35[/tex]
Inverse operations,
[tex]7x+4=35[/tex]
[tex]7x=31[/tex]
[tex]x=\frac{31}{7}[/tex]
find the HCF of the following numbers by listing the set of factors class six the question is 60 and 90
Answer:
30
Step-by-step explanation:
If P =
1 2
then, prove that P2 - 2P - 5 = 0, 1 = 0, where I and 0 are unit
3 1
matrix and null matrix of order 2 x 2 respectively.
Answer:
thanks for free ponits you dint gave it free but I theft I'm sorry
is 10 X -3+ 2X -5 and four (3X -2) equivalent?
Answer:
Yes, both equations are equivalent
Step-by-step explanation:
Simplifying [tex]10x - 3 + 2x - 5[/tex] by combining "like terms" gives us [tex]12x-8[/tex].
Now, to solve "four (3X - 2)", we need to use the distributive property. Doing [tex](4\cdot3x)-(4\cdot2)[/tex] gives us [tex]12x-8[/tex] again.
Therefore, [tex]10x - 3 + 2x - 5[/tex] and four (3X -2) are equivalent!
(Hope this helped you! If any part of the explanation doesn't make sense, or the answer is incorrect/not the answer you we're looking for, then please tell me and I will try to fix my answer as soon as possible)
helphelphelphelphelphelphelp
Answer:
P = 1,-10
Q=1,-1
R=7,-1
S=7,-10
1. The product of two consecutive numbers is 42. What are the numbers?
Answer:
6, 7
Step-by-step explanation:
6 x 7 = 42
Write an equation that represents the statement "the
product of a number, x, and the number 7 is 42."
Answer:
7x = 42
Step-by-step explanation:
"Product" refers to multiplication and "is" refers to equal to.
Hi! I'm happy to help!
This equation will be written like this
x×7=42
To make this easier to solve, we can use the inverse operation, division.
42÷7=x
42 divided by 7 is 6, so the answer is 6.
I hope this was helpful, keep learning! :D
HELP FAST! 100 POINTS
Answer:
F(8,6)
Step-by-step explanation:
(x₁,x₂) --> E (-7, 2)
F(x₂,y₂)
Midpoint M(0.5 , 4)
Midpoint = [tex](\frac{x_{1}+x_{2}}{2},\frac{+y_{1}+y_{2}}{2})[/tex]
[tex](\frac{-7+x_{2}}{2},\frac{2+y_{2}}{2}) = (0.5, 4)\\\\[/tex]
Compare x,coordinates and y-coordinates,
[tex]\frac{-7+x_{2}}{2}= 0.5 \ ; \ \frac{2+y_{2}}{2}=4\\\\[/tex]
-7 + x₂ = 0.5*2 ; 2+y₂ = 4*2
-7+ x₂ = 1 ; 2 +y₂ = 8
x₂ = 1+ 7 ; y₂ = 8 - 2
x₂ = 8 ; y₂ = 6
F(8,6)
Answer:
F(8,6)
Step-by-step explanation:
(x₁,x₂) --> E (-7, 2)
F(x₂,y₂)
Midpoint M(0.5 , 4)
Midpoint =
Compare x,coordinates and y-coordinates,
-7 + x₂ = 0.5*2 ; 2+y₂ = 4*2
-7+ x₂ = 1 ; 2 +y₂ = 8
x₂ = 1+ 7 ; y₂ = 8 - 2
x₂ = 8 ; y₂ = 6
F(8,6)
find the surface area of the prism.
Step-by-step explanation:
area of the base is the area of the rectangle 5×12.
B = 5×12 = 60 in²
the height h is then the side going up from that base rectangle : 4 in.
the perimeter of the base (rectangle) is 2×length + 2×width
P = 2×12 + 2×5 = 24 + 10 = 34 in.
HELP PLS. Team A built 8 sandcastles in 2 hours and Team B built 5 sandcastles in 1 hour. Which team worked at a faster rate?
Answer:
[tex]team \: a \\ 8 \: in \: number \: with \: in \: 2hours \: \\ so \: in \: 1hour \frac{8}{2} = 4 \\ \\ team \: b \\ 5 \: in \: number \: in \: 1 \: hour \\ so \: b \: is \: fast \\ thank \: you[/tex]
a ceartain trigangle has to 45 degree angles . what type of triangle is it
Answer:
Either a 45°-45°-90° triangle or an isosceles right triangle depending on the question
Step-by-step explanation:
A triangle that has two 45° angles always has a third angle of 90° making it a 45-45-90 as well as a right triangle.
Please help me out with math will mark like brainlist if answers ASAP
Answer:
13/16
Step-by-step explanation:
To add these two fractions multiply the first fraction by 2 to get equivalent denominators so 6/16+7/16 then add the numerators to get 13/16.
Answer:
13/16
Step-by-step explanation:
3/8 + 7/16
Get a common denominator
3/8*2/2 + 7/16
6/16 + 7/16
13/16
4. If triangle ABC has the following measurements, find the measure of side c:
a = 5
B = 7
C = 42 degrees
9514 1404 393
Answer:
c ≈ 4.69
Step-by-step explanation:
The Law of Cosines is used for the purpose. It tells you ...
c² = a² +b² -2ab·cos(C)
c² = 5² +7² -2·5·7·cos(42°) ≈ 21.9799
c ≈ √21.9799 ≈ 4.688
The measure of side c is about 4.69 units.
how do I answer these questions?
Answer:
cross multiplication would be the quickest technique that works to visualize this set of questions..
if [tex]\frac{a}{b} = \frac{c}{d}[/tex] then ad = bc
each item that does not have a denominator add a "1" as the denominator
so first item... [tex]\frac{sin(50)}{1} = \frac{x}{4}[/tex]
[tex]\frac{4 * sin(50) }{1} = x[/tex] x = [tex]4\sin \left(50^{\circ \:}\right)=3.06417\dots[/tex]
#b) [tex]3\tan \left(81^{\circ \:}\right)=18.94125[/tex]
#c) [tex]6\cos \left(33^{\circ \:}\right)=5.03202[/tex]
.
.
#g)[tex]7.1\tan \left(18.4^{\circ \:}\right)=2.36185[/tex]
Step-by-step explanation:
Answer:
First we extract the sin or the tan according to the question and multiply the number with the denominator and extract the x .
a)[tex] \sin(50) = \frac{x}{4} \\ 0.766 = \frac{x}{4} \\ x = 3.06[/tex]
b)[tex] \tan(81) = \frac{x}{3} \\ 6.31 = \frac{x}{3} \\ x = 18.93[/tex]
C)[tex] \cos(33) = \frac{x}{6} \\ 0.83 = \frac{x}{6} \\ x = 0.83 \times 6 \\ x = 4.98[/tex]
d)[tex] \cos(75 ) = \frac{x}{3.5} \\ 0.258 = \frac{x}{3.5} \\ x = 0.2588 \times 3.5 \\ x = 0.90[/tex]
e)[tex] \sin(24) = \frac{x}{4.2} \\ 0.406 = \frac{x}{4.2} \\ x = 0.406 \times 4.2 \\ x = 1.70 [/tex]
f)[tex] \tan(42) = \frac{x}{10} \\ 0.9 = \frac{x}{10} \\ x = 9[/tex]
g)[tex] \frac{x}{7.1} = \tan(18.4) \\ \frac{x}{7.1} = 0.33 \\ x = 0.33 \times 7.1 \\ x = 2.34[/tex]
h)[tex] \frac{x}{5.3} = \sin(64.7) \\ \frac{x}{5.3} = 0.9 \\ x = 4.77[/tex]
i)[tex] \frac{x}{12.6} = \cos(52.9) \\ \frac{x}{12.6} = 0.6 \\ x = 0.6 \times 12.6 \\ x = 7.56[/tex]
I hope I helped you^_^
Is the quotient of a rational number and
an irrational number always irrational?
Explain.
HELP ASAP
Answer = No,It can be either rational or irrational.
________________________________
Prroof —
4 ÷ √8 = √2 { Which is an irrational number }
________________________________
Now, Here comes a twist,
We know that — 0 is a rational number.
And, 0 ÷ √56666 = 0 ________________________________
Note—
But, √56666 ÷ 0 = Error ( Lol )
________________________________
which of the following represents…
Algebra 1 need help ASAP
Answer:
It's A
Step-by-step explanation:
The money earned is 160 in 7 hours
The money earned is 100 in 3 hours
The slope = change in money / change in time
The slope = (160 - 100 ) / (7 - 3) = 60/4 = 15 dollars / hour
The slope means that for every hour he earns 15 dollars.
tìm khoảng đồng biến, nghịch biến của hàm số
y= -5x+3
Answer:
sgehwjwnnwjwjj2629937463u83
