
Math
Given the geometric series : 1/4 + x + 1/36 + 1/108+ .... We need to find the value of x. Let (r) be the common difference between terms. Then we know that: x = 1/4 * r...........(1) 1/36 = x *...

Math
First we will recall the formula of the volume of the cone. We know that the volume if given by the equation: V = (1/3)*r^2*pi * h where r is the radius and h is the height. Now, given that V = 142...

Math
Given the equations: x*y = 144.............(1) x+ y= 30 .............(2) We will use the substitution method to find the values of x and y. from (2) we know that y= 30x Now we will substitute into...

Math
Given the numbers 42, 126, and 210 We need to find the greatest common factor ( GCF) First we will factor each numbers. ==> 42 = 2*3*7 ==> 126 = 2*3*3*7 ==> 210 = 2*3*5*7 Now we will...

Math
Given the equations: 3x^2  2y 2x^2  3y Also we know that x= 3 and y= 5 We need to know the difference in values between both equations. First we will substitute x and y values in each equation....

Math
The car travels 27 mile / gallon. We need to calculate the cost for the car to travel 2727 miles. First we need to determine how many gallons does the car needs to travel 2727 miles. We know that:...

Math
The complex number is written in the form z = a+ bi. We need to determine the result of the product of two complex numbers. Let z1= a+ bi and z2 = c+di Now we need to calculate the product of...

Math
The area of the rectangle is 2.9 square meters. We know that the perimeter is 2l + 2w if l is length and w is width. We know that l = w + .8 So 2l + 2w = 7 2 (w + .8) + 2w = 7 2w + 1.6 + 2w = 7 4w...

Math
If (2n+1)/n is a natural number, then 2n+1 is divisible by n. Then we will rewrite as follow. 2n+1)/n = k where k is a natural number. ==> (2n/n + 1/n ) = k ==> 2 + 1/n = k Now since k is a...

Math
1/100 + 2/100 + 3/100 + ....+ 99/100 Since the denominators are the same, then we will add the numerators. ==> (1+2+3+...+99)/100 Now we notice that the numerator is a sum of a series Then we...

Math
ctgx + cosx = 1+ ctg*cosx First we will rewrite the identities. We know that: ctg(x) = cosx/sinx. ==> cosx/sinx + cosx = 1 + cosx/sinx * cosx Now we will simplify. ==> (cosx + cosx*sinx)/sinx...

Math
4sinx*cosx 1 = 2(sinxcosx) We will expand the brackets. ==> 4sinxcosx 1 = 2sinx 2cosx Now we will move all terms to the left side. ==> 2cosx 2sinx +4sinxcosx 1 = 0 ==> 2cosx...

Math
We can calculate the monthly payment for a mortgage using the formula: M = P [ i*(1 + i)^n] / [ ((1 + i)^n)1] where M is the monthly payment, i is the annual rate of interest divided by 12, n...

Math
The function for the profit is given as: P(x)=(5x400)/(x+600) Note: Here, the domain is x > 0. I have taken this as the domain under the assumption that you cannot sell less than 0 kilograms of...

Math
The function f(x) = (3x^0)  x^(1/2) Now any number to the power 0 is 1 So f(x) = 3  x^(1/2) f(9) = 3  9^(1/2) => 3  3 => 0 Therefore f(9) = 0

Math
The picture below represents the graph of the function `y = 4^x` . b) You need to reflect the graph over the x axis such that: Notice that the red curve is the graph of original function `y =...

Math
We have to solve 3^(x^2+4x)=(1/9)^2 for x. 3^(x^2+4x)=(1/9)^2 => 3^(x^2+4x)=(1/3^2)^2 => 3^(x^2+4x)=(3^2)^2 => 3^(x^2+4x)= 3^4 As the base 3 is the same, we can equate the exponent....

Math
The monthly payment for a mortgage can be calculated using the formula: M = P [ i*(1 + i)^n] / [ ((1 + i)^n)1], where M is the monthly payment, i is the annual rate of interest divided by 12, n...

Math
Starting with y = 2^x, it is not possible to get the result y = 3*log(3) (125*x^3) We can achieve the same by the following: y = 3*log(3) (125*x^3) Use log b^a = a*log b => y = 3*log (3)...

Math
We will draw a line between the point (3, 4) and the origin point. Then angle x located between the line and the xaxis. Then, we hace formed a right angle triangle where the base = 3, the height...

Math
y= x^2 + 3x + 18 We need to find the xintercept of the curve y. The xintercept is the point where the curve y meets the xaxis. Then the values of y would be zero. ==> x^2 + 3x + 18 = 0...

Math
Let f(x) = x/(x+2) + 3/(x4) To find the roots, we will rewrite as one fraction. ==> f(x)= [ x(x4) + 3(x+2) ] / (x+2)(x4) ==> f(x) = ( x^2  4x + 3x + 6) / (x+2)(x4) ==> f(x) = (x^2 x...

Math
The roots of the quadratic equation are x= 3 and x = 5 So we can write: (x  3)(x  5) = 0 => x^2  3x  5x + 15 = 0 => x^2  8x + 15 = 0 The required quadratic equation is x^2  8x + 15 = 0

Math
We are given that x^2+ kx  6 = (x  2)(x + 3) Now x^2+ kx  6 = (x  2)(x + 3) => x^2 + kx  6 = x^2 2x + 3x  6 => x^2 + kx  6 = x^2 + x  6 canceling the common terms => kx = x =>...

Math
Given the curve f(x) = 3x^2 + 9x. We need to find the extreme value of the function. First we notice that the coefficient of x^2 is negative, then the curve will have a maximum point. Now we will...

Math
Given the fraction (8x+14)/ (x+1)(x+5) We need to rewrite into the form A/(x+1) + B/(x+5) ==> (8x+14)/ (x+1)(x+5) = A/(x+1) + B/(x+5) We will rewrite with the common denominator. ==>...

Math
We have to write the polynomial P(x) = 5x^3 2x^2 + 5x  2 as a product of linear factors. P(x) = 5x^3 2x^2 + 5x  2 => P(x) = x^2( 5x  2) + 1( 5x  2) => P(x) = (x^2 + 1)(5x  2) The term...

Math
Given the polynomial P(x) = x (x2)^2 (x+2) We need to determine the zeros of P(x). We notice that P(x) is already written in its factors form. Then the zeros of P(x) are the zeros of the...

Math
The degree of a polynomial is the largest exponent when the polynomial is written in standard form. So, expand this polynomial into standard form: 5(x  2)(x3 + 5) + x5 = x^5+5 x^410 x^3+25 x50...

Math
Given that f(x) is a function such that its zeros are 0, 2, 3. Then we know that the factors are ( x) ( x+2) and (x3) Then we will write the function as follows: f(x) = x (x+2) (x3) Now we need...

Math
We need to determine the value of 20% of 2. First we will rewrite the percent as a fraction of 100. We know that A% = A/100 ==> 20% = 20/100 Now let us simplify the fraction by dividing both...

Math
Given the equations: ax + by = c bx  ay = c We notice that the above are equations of two lines. We need to find the relation between the lines (i.e perpendicular, parallel, or neither). To...

Math
RATION is a 6 letter word with 3 vowels. As the 3 vowels have to appear together, there are 4 elements with one element made up by the three vowels together and the other 3 consonants. The 4...

Math
Given that log(4) x = 12 We need to find log(2) x/4 First we will use logarithm properties to simplify. We know that: log a/b = log a  log b ==> log2 (x/4) = log2 x  log2 4 Now we will rewrite...

Math
The probability is 1/2 because there are only two outcomes: heads or tails. Since it is equally likely that either a heads or a tails will result from a coin flip, this means that the probability...

Math
We have to find x from log(x) 1/8 = 3/2 log(x) 1/8 = 3/2 => log (x) 8^1 = 3/2 => log (x) 4^(3/2) = 3/2 take the antilog of both the sides => 4^(3/2) = x^(3/2) as the exponent is...

Math
The probability that the boy throws the ball in the box is given as 1/4. He is given 4 attempts and we need to find the probability that he will throw the ball in the box at least once. The...

Math
These sets are equal. They are equal because every element of the first set is also an element of the second set. The elements in the second set are expressed differently, than the elements of the...

Math
Given the interval 3 < a < 5 We need to find the opposite interval for a. Let us rewrite as interval. ==> a belongs to (3, 5) Now to determine the opposite we consider that a is a real...

Math
Let E =3*3*4+732 First thing, we need to rewrite the absolute values in the form of a real number. Let us simplify. l3 l = 3 l3l = 3 l4l = 4 l 7 l = 7 l3l = 3 l2l = 2 Now we...

Math
x+24=10 x+y+24=10 Let us solve the system and determine if we have a solution in the set of natural number. We will use the elimination method. We will subtract (1) from (2). ==> y = 0 Now to...

Math
The radical numbers ar numbers written into the form of sqrt(a). There are rules for simplifying. Rule (1): sqrt(a*b) = sqrt(a)* sqrt(b) sqrt(a/b) = sqrt(a)/sqrt(b) Example: sqrt15 = sqrt(3*5) =...

Math
Equality is the mathematical term for the equations, such that the left side and the right side are equal. We describe the equality with the symbol " = " The statement is that the right side...

Math
An integer is any whole number. That is, it is a number that has no fractional or decimal component attached to it. For example, the number 1.5 is not an integer because it has a decimal...

Math
Yes, you have enough. The slope intercept form of the equation of a line is y = mx + b Right now, you have everything but "b." So all you have to do is plug numbers in and solve for b. 1 = 2(1)...

Math
This depends on what information you already have. The easiest way to find this is if you are given the coordinates of two points on the line. Assuming they are in (x,y) form, you can use the...

Math
The mistake here is that the line y = 5 is not vertical. The y axis is vertical, but this line would be horizontal. Therefore, they would be perpendicular. If the line is y = 5, then y = 5 for...

Math
x  y = 3 Therefore  y = 3  x Multiply both sides by 1 y = 3 + x Substitute this in to the other equation. 2x + 3 + x = 12 Add the x terms 3x + 3 = 12 Add 3 to both sides 3x = 15 Divide...

Math
One way to do this is if you can express one of the two variables in terms of the other variable. So, for example, if you have a problem with variables x and y, you can write it just using x if you...

Math
First we will rewrite the intervals as inequalities. ==> x E [4, inf) ==> x >= 4 We will multiply by 1 and reverse. ==> x =< 4 Now subtract 0.5 from both sides. ==> x 0.5...