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You're almost there.
I can see you are setting the denominator to 0 and you have:

Using Quadratic formula: for the equation
We have a = 1 , b=10, c =25

or
or
or x =-5
This shows that denominator becomes 0 for x= 5.
Therefore, we can say that x =-5 can not be in the domain of the given fucntion.
Hence, except for x =-5, all real numbers are the domain.
It is represented as:
x ϵ R - {-5}
or x ϵ (-∞,∞) - {5}

Hope this helps~~

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Distance to be traveled = 20 miles
Speed at which you're traveling = 70 miles per hour = 70 miles/ 60 minutes
= 7/6 miles per minute
Let the time taken by you to reach boulder city be t.
We know that Speed =
So, 7/6 miles per minute = 20 miles/t
7*t = 6*20 {Cross multiplication}
7t = 120
t = 120/7
or t = 17.14 minutes
Hence, it will take 17.14 minutes for you to reach there.

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Let the ages of Tom and Alice be x and y respectively.
We know that Tom's age is thrice as Alice's age.
So, x = 3y --------------- (1)
Also, we know that the sum of their ages is 54.
Therefore,
x + y = 54
or 3y + y = 54 { Using equation (1) }
4y = 54
or y = 54/4 = 13.5
Plugging back the value of y in equation (1), we get:
x = 3*13.5 = 40.5
Hence, Tom's age is 40.5 years and Alice's age is 13.5 years


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We know that side length of a cube is 8 inches.
Volume of the cube =
V(cube) =
V(cube) = 512 cubic inches
WE know that the radius of the cylinder is r = 4 inches and let the height of the cylinder be h inches.
Volume of the cylinder = π
V(cylinder) =
V(cylinder) = 16πh
Since the volume of cube and cylinder are same, so
V(cylinder) = V(cube)
or 16πh = 512
or πh = 512/16 {Dividing both the sides by 16}
or πh = 32
or h = 32/π inches
If we wish to get the decimal value fo h, we can use the standard value of π = 3.14
Therefore, h = 32/π = 32/3.14
or h = 10.189
or h = 10.2 inches (approximately)


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Let the ages of Tom and Alice be x and y respectively.
We know that Tom's age is thrice as Alice's age.
So, x = 3y --------------- (1)
Also, we know that the sum of their ages is 54.
Therefore,
x + y = 54
or 3y + y = 54 { Using equation (1) }
4y = 54
or y = 54/4 = 13.5
Plugging back the value of y in equation (1), we get:
x = 3*13.5 = 40.5
Hence, Tom's age is 40.5 years and Alice's age is 13.5 years
Hope this helps~!

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Let the number be x.
Difference of the number and 5 = x -5
2 times the difference of the number and 5 = 2(x - 5)
3 times the difference of the number and 5 = 3(x - 5)
Sum of 2 times and 3 times the difference of the number and 5 = 2(x - 5)+ 3(x-5)
= 5( x- 5)
Difference of the number and 1 is = x -1
4 times the difference of the number and 1 is = 4 (x -1)
So, 5 (x -5) = 4(x -1)
or 5x - 25 = 4x - 4
or 5x - 4x - 25 = -4
or x - 25 = -4
or x = 25 -4
or x = 21
Hence, the number is 21

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Let the positive integer be x.
Reciprocal of the integer would be =
Now, we can set up the equation as:

or {Multiplying both the sides by x}


{ Factoring by regrouping}



Since we have assumed that the integer is positive or x > 0
Therefore, x =4
Hope this helps~!

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÷
Rule: Whenever we have two fractions divided, then we use "KSS" = Keep Switch Switch.
Let me explain this to you a little. We 'keep' the first fraction, switch the operator from division to multiplication and again we switch the second fraction.
=
= { Cancelling out x+1 from both the fractions }
=
=
Hope this helps~!

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Given equation of line is: 3x + 2y = 5
Let us transform this into slope intercept form: y = mx + b
2y = 5 - 3x
or y = -3x/2 + 5/2
Comparing it with slope-intercept form: y= mx + b
Slope = m = -3/2
If two lines are parallel, then they have equal slopes.
So, the slope of the line parallel to 3x + 2y = 5 is -3/2
Using slope-intercept form, we can write the equation of a new line can be written as:
y = mx + b
y = -3x/2 + b
Since this line also passes through (-2,-8)
Therefore,
-8 = (-3)*(-2)/2 + b
-8 = 3 + b
b = -8 -3
b = -11
Hence the equation of the line parallel to 3x + 2y =5 is:
y = -3x/2 - 11
or 2y = - 3x - 22 { Multiplying both the sides by 2}
or 3x + 2y + 22 = 0

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Let the ages of Mary and John be x and y years respectively today.
3 years ago, age of John was (y-3) years
Since Mary is twice as old as john was 3 years ago.
So, x = 2(y-3)
or x = 2y - 6 -------- (1)
In 5 years, age of John will be = (y + 5) years
"John will be 1 year older than Mary is now"
So, (y+5) = x + 1
or x = y + 5 -1
or x = y + 4 --------(2)
Using equation (1) and (2), we get:
2y - 6 = y + 4
2y = y + 4 + 6
2y = y + 10
y = 10

Plugging back the value of y in equation (2)
x = 10 + 4
x = 14

Hence, Age of Mary today is x= 14 years and age of John today is y=10 years.
Hope this helps~!

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Let the speed of the wind be w miles per hour.
Let t be the time taken by pilot to travel 560 miles with the wind and also to travel 360 miles against the wind.
Case: When traveling with the wind,
total speed of pilot = speed of pilot in still air + speed of wind
Vt = 230 + w
Distance traveled = total speed * time taken
560 = (230 + w)* t
------------ (1)

Case: When traveling against the wind,
total speed of pilot = speed of pilot in still air - speed of wind
Vt = 230 - w
Distance traveled = total speed * time taken
360 = (230 - w)* t
------------- (2)

Using equation (1) and (2),

{ Dividing both the sides by 40 }
{ Cross-multiplication}
3220 - 14w = 2070 + 9w
3220 - 2070 = 14w + 9w
1150 = 23w
w = 1150/23
w = 50
Hence, the speed of the wind is 50 miles per hour.
Hope this helps~!

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Let the two angles be x and y such that x>y.
Since they are supplementary angles, therefore x + y = 180 ---------(1)
Also, x = y + 46 ------------ (2)
Plugging the value of x from equation (2) into equation (1), we get:
y + 46 + y = 180
2y + 46 =180
2y = 180 - 46
2y = 134
y = 67
Plugging back the value of y in equation (1),
x = 67 + 46
x = 113
Hence, the two angles are 67' and 113'

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Given that the volume of a box is 157500 cm^3 and base area is 45 * 50 cm^2
We know that volume formula of a box is
Let the height of the box be h cm.
So, 157500 =45*50*h
157500 = 2250*h
or

h = 70 cm
Hence the height of the box is 70 cm.
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a1 = 10, r =-1
nth term of a geometric sequence is given by:
So, a2 = a1*r^(2-1) = a1*r = 10*(-1) = -10
a3 = a1*r^(3-1) = a1*r^2 = 10*(-1)^2 = 10*1 = 10
a4 = a1*r^(4-1) = a1*r^3 = 10*(-1)^3 = 10*(-1) = -10
a5 = a1*r^(5-1) = a1*r^4 = 10*(-1)^4 = 10*1 = 10
Hope this helps~!

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Let the other sides of the right triangle be x and y.
Since the length of the wire was 60 cm, the whole 60 cm has been bent to form a right triangle and that means 60 cm is as well the perimeter of the triangle.
Therefore, x + y + 26 = 60
or x + y = 34
or y = 34 -x ---------------- (1)
Using the Pythagoras theorem,
x^2 + y^2 = 26^2
x^2 + (34 - x)^2 = 26^2 {Using equation (1)}
x^2 + 34^2 + x^2 - 68x = 26^2
2x^2 - 68x + 34^2 - 26^2 = 0
2x^2 - 68x + (34 -26)(34+26) = 0 { a^2 - b^2 = (a+b)(a-b)}
2x^2 - 68x + 8*60 = 0
2x^2 - 68x + 480 = 0
x^2 - 34x + 240 = 0 { Dividing the equation throughout by 2}
x^2 - 24x - 10x + 240 = 0
x (x -24) - 10(x-24) =0
(x -10) (x-24) = 0
x =10, x =24
Plugging back the value of x in equation (1), we get:
When x =10, y = 34 -10 = 24
When x= 24, y = 34-24 =10
Hence, the two legs of the right triangle are 10 cm and 24 cm.

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Let the present age of father be x and the son be y.
Now, x = 4y ---------- (1)
After 16 years,

age of father will be x+ 16
age of son will be y +16
Therefore, x + 16 = 2( y + 16)
x + 16 = 2y + 32
x = 2y + 16
or 4y = 2y + 16 {Using equation (a)}
2y = 16
y = 8
Plugging back the value of y in equation (1), we get:
x = 4*8 = 32
Hence, the present age of father is 32 years and present age of son is 8 years.

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Let the number be x.
So, we can set up the equation as:
(3/4)x = (1/3)x + 15
Solving for x:
3x = (4/3)x + 60 {Multiplying 4 on both the sides}
9x = 4x + 180 {Multiplying 3 on both the sides}
5x = 180
x = 36
Hope this helps~!

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A computer can perform 30 identical tasks in 6 hours.
Rate at which a computer works = number of tasks/ number of hours = 30/6 = 5 tasks per hour
We can say that a computer performs 5 tasks in an hour.
So, a computer performs 3*5 = 15 tasks in 3 hours.
Now, if x number of computers are operated for 3 hours. Then they can do number of tasks = number of tasks done by a computer in 3 hours times the number of computers
Therefore, Number of tasks done by x computers in 3 hours = 15*x = 15x
Now, this number of tasks should equal 80 as mentioned in the problem.
Therefore, 15x = 80
x = 80/15
x = 16/3
x = 5 and 1/3
Oh we can't have fraction number of computers.
Hence, we can say to complete 80 tasks, there should be 6 computers operating simultaneously.

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7/n = 8/7
Cross-multiplication: Denominator of left side fraction gets multiplied with the numerator of the right side fraction and simultaneously, the denominator of the right side fraction gets multiplied with the numerator of the left side fraction.

So, 7*7 = n*8
or 49 = 8n
Therefore, n = 49/8

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Let the measures of equal sides be x and the measure of the shortest side be y.
Also, it is given that the measure of equal sides is double of the shortest side.
So, x = 2y ----------- (1)
We know that perimeter of triangle is sum of all the three sides.
Therefore, x + x + y = 36
or 2y + 2y + y = 36 {Using equation (1) x = 2y}
5y = 36
y = 36/5
Using equation (1),
x = 2y = 2*(36/5) = 72/5
Hence, the sides of triangle are 72/5, 72/5 and 36/5

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Let the distance walked be x.
The total course would be 1 (1/1)
So, distance ran + distance walked = total course length
3/5 + x = 1
x = 1 - 3/5
x = 5/5 - 3/5
x = (5-3)/5
x = 2/5
She walked 2/5 of the course.

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Let x and y be the two numbers whose sum is 100 such that x>y.
So, x + y = 100 ------------- (1)
Also, larger number is 10 more than twice the smaller.
Therefore, x = 2y + 10
100 - y = 2y + 10 {Using equation (1) }
90 = 3y
Dividing both the sides by 3, we get:
90/3 = y
or y = 30
Plugging back the value of y in equation (1), we get:
x = 100 - y
x = 100 -30
x = 70
Thus, the larger number is 70 and the smaller number is 30.

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4^x = 0.5
Applying ln on both the sides, we get:
ln (4^x) = ln(0.5)
x ln4 = ln(0.5)
Dividing both the sides by ln4
x = ln(0.5)/ln4
x = -0.693147/1.386294 {Using calculator}
x = - 0.500000
or x = -1/2 {Converting desimal to a fraction}

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Let that number be x.
So, we can write:
x times 6 = 30
x *6 = 30
Dividing both sides by 6, we get:
x = 30/6
x = 5
Hence that number is 5.

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2log3x = log3 (x+6)
Using the property of logarithm such that n log a = log a^n
log3 x^2 = log3 (x+6)
Since the logs on both the sides have same bases, they can be canceled out.
So, x^2 = x + 6
x^2 - x - 6 =0
Now, factoring by regrouping:

x^2 - 3x + 2x -6 = 0
x(x -3) + 2(x-3) = 0
(x-3) (x +2) = 0
x = 3 , x =-2
Since originally we had the term 2 log3 x..which shows that x should always be greater than 0 {Because log of negative number is not defined}
Thus, x = 3
Hope this helps~!

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Hello there, I see what you have been missing. I will provide the step by step solution below:
sqrt(a^2 + b^2) = c
sqrt( (4y)^2 + y^2 ) = 20.6
sqrt( 16y^2 + y^2) = 20.6
sqrt(17 y^2 ) = 20.6
Squaring on both the sides, we get:
17 y^2 = 424.36
Dividing both the sides by 17, we get:
y^2 = 424.36/17
y^2 = 24.96235294117647
y = 4.996233875748459 , y = - 4.996233875748459
Since y is the length of one of the sides, it can not hold a negative value.
Thus, y = 4.996233875748459
One can round off the answer to 3 decimal places as: y = 4.996