
Math
The logarithmic function is defined such that if log(a) b = c, we have b = a^c. The equation to be solved is log(4) (x  2) = 1 => x  2 = 4^(1) => x  2 = 1/4 => x = 2 + 1/4 => x =...

Math
The graph of the function of the form y = a*x^2 becomes more steep as the value of a in increases. For the same increase in x the value of y changes by a larger extent as x is being multiplied by a...

Math
Let the number we need to find be x and y. The sum of the numbers is x + y = 720...(1) We also know that one number is half the other. x = y/2 substitute y = 2x in equation (1) x + y = 720 => x...

Math
We have z = x  iy and w = y  ix We need the values of z that satisfy z^2 = w z^2 = (x  iy)^2 = x^2  y^2  2xyi = w So we have x^2  y^2  2xyi = y  ix equate the real and complex coefficients...

Math
We have to prove that : (1+ sin 2x) / sin^2 x = csc^2 x + 2cot x Start from the left hand side: (1+ sin 2x) / sin^2 x use the relation sin 2x = 2*sin x*cos x => (1 + 2*sin x*cos x) / (sin x)^2...

Math
We have to prove that: (1  sec(x))/(1 + sec(x)) = (cos(x)  1)/(cos(x) + 1) Let's start from the left hand side (1  sec(x))/(1 + sec(x)) use the fact that sec x = 1/cos x => (1  (1/cos x))/(1...

Math
We have to solve the system of equations: 3x + 4y = 38…(1) 5x + 5y = 30…(2) (2)*(3/5) => 3x + 3y = 18 Subtract this from (1) => 3x + 4y – 3x – 3y = 38 + 18 => y = 56 substitute y...

Math
The vertical asymptotes are vertical lines that pass through the zeros of the denominator. Here, we have [2x^2 + x + 1] / [x + 2], We can see that equating the factors of the denominator to 0 gives...

Math
The integral of f(x)=(x+1)/(x^2+2x) can be found using substitution. If y = x^2 + 2x dy/dx = 2x + 2 = 2(x + 1) => (1/2)dy = (x + 1)dx Int[(x+1)/(x^2+2x) dx] => Int[(1/2)*(1/y) dy] =>...

Math
It is given that f(x)=2x+1 and g(x)=x+2. We need to solve the equation gogog(x)=fofof(x) gogog(x) => gog(x + 2) => g(x + 2 + 2) => g(x + 4) => x + 6 fofof(x) => fof(2x + 1) =>...

Math
x2y = 11............(1) x + 5y = 23...........(2) We will use the substitution method to solve the system. First we will rewrite equation (1). ==> x = 2y11 Now we will substitute into...

Math
We need to find x for lg(x)+lg(x+1)=(lg90) to exist. Lets simplify the expression: lg(x) + lg(x+1) = lg(x*(x + 1)) = (lg 90) x(x + 1) = 90 x^2 + x = 90 => x^2 + x  90 = 0 => x^2 + 10x  9x ...

Math
We need to solve x=1+square root(1+square root x) x = 1 + sqrt (1 + sqrt x) => x  1 = sqrt (1 + sqrt x) square the sides => x^2 + 1  2x = 1 + sqrt x => x^2  2x = sqrt x square the...

Math
We know that Pn(x)=(x1)(x2)...(xn). We have to solve the inequality P1(x)/P2(x)>=2 P1(x) = (x 1) P2(x) = (x  1)(x  2) P1(x) / P2(x) = (x 1)/(x  1)(x  2) >=2 => (x  1)>=2*(x ...

Math
We need to find the integral of y = 6x^5 + 2x  1. Use the property that the integral for x^n is (1/(n + 1))*x^(n + 1) for each of the terms. Int[6x^5 + 2x  1 dx] => 6*x^6 / 6 + 2*x^2/2  x/1...

Math
We'll write the sum: Sum k(k + 3), where k is an integer number whose values are from 1 to n. Sum k(k+3) = Sum (k^2 + 3k) Sum (k^2 + 3k) = Sum k^2 + Sum 3k Sum k^2 = 1^2 + 2^2 + ... + n^2 =...

Math
You need to use the following derivative rules, such that, power rule, quotient rule and chain rule, in this order. `f'(x) = (1/2)((cos2x)/(x^2+x+1))^(1/2  1)*((cos2x)'(x^2+x+1) ...

Math
We have to find the roots of 9x^2  6x + 21 = 0 by completing the square. 9x^2  6x + 21 = 0 => (3x)^2  2*3x + 1 = 20 => (3x  1)^2 = 20 => 3x  1= sqrt 20 and 3x + 1 = sqrt (20)...

Math
We have to find the roots of (4^2)^x3*4^x+2=0 (4^2)^x3*4^x+2=0 Let 4^x = y => y^2  3y + 2 = 0 => y^2  2y  y + 2 = 0 => y(y  2)  1(y  2) = 0 => (y  1)(y  2) = 0 => y = 1...

Math
A sequence is convergent if the sum of n terms of the sequence approaches a constant value as n tends to infinity. Here, we have the sequence made up of 1, 1, 1, 1, 1, 1... If we add up the...

Math
Trigonometry is the branch of mathematics that deals with the relation between the angles of a triangle and its sides. The functions in trigonometry find application in defining periodic functions...

Math
We notice that 0 <= fn(x) <= x^2n if x belongs to the range [0,1). We'll integrate and we'll get : 0 <= int fn(x)dx <= int x^2ndx We'll calculate int x^2ndx=x^(2n+1)/(2n+1) We'll apply...

Math
We have to find the second derivative of f(x) = (x^3+3x)*(3x^26x) + 9x^2/(x1) f(x) = (x^3+3x)*(3x^26x) + 9x^2/(x1) f'(x) = (3x^2 + 3)(3x^2  6x) + (x^3 + 3x)(6x  6) + 18x/(x  1)  9x^2/(x ...

Math
We have x o y = xy  x  y + 2 and f(x) = x + 1. f(xy) = xy + 1 f(x) o f(y) = (x + 1) o (y + 1) => (x + 1)(y + 1)  (x + 1)  (y + 1) + 2 => xy + x + y + 1  x  1  y  1 + 2 => xy + 1 We...

Math
We have to find the derivative of f(x) = e^[x/(x+2)] We use the chain rule here: f'(x) = e^[x/(x + 2)]*[(1/(x + 2)  x/(x + 2)^2] => e^[x/(x + 2)]*[((x + 2  x)/(x + 2)^2] => e^[x/(x +...

Math
We have to differentiate y = ln(1 + 3x + 4x^2) We use the chain rule here. y = ln(1 + 3x + 4x^2) y' = [1/(1 + 3x + 4x^2)]*(3 + 8x) The required derivative of y = ln(1 + 3x + 4x^2) is [1/(1 + 3x +...

Math
You may also use reminder theorem, such that: `p(x) = (x  1)^2*q(x) + r(x)` The problem provides the information that the polynomial `p(x)` is exactly divided by `(x  1)^2` , hence, `r(x) = 0.`...

Math
You need to solve for x the following irrational equation, such that: `x  a = sqrt(x^2  1)` The problem provides the information that `a = 0` , such that: `x = sqrt(x^2  1) => x^2 = (...

Math
We are given that 25x^2+a+36y^2 and 9x^4/25b+25x^2/9 are squares and we need to determine a and b. The two can be squares of the form (a  b)^2 and (a + b)^2 25x^2+a+36y^2 = (5x)^2 + (6y)^2 + a a...

Math
We need to find the value of ad  bc if a,b,c,d are the terms of a geometric series. As a, b, c and d are terms of a geometric series. b=a*r, c=a*r^2 and d = a*r^3 a*d = a^2*r^3 and b*c = a^2r^3 We...

Math
We have the function f(x) = (2x+4/3x1)^3. The derivative f'(x) can be found using the chain rule. f'(x) = 3*(2x+4/3x1)^2*(2x+4/3x1)' f'(x) = 3*(2x+4/3x1)^2*((2x+4)*(3x  1)^2*3 + (3x ...

Math
The partial fractions can be found in a very easy way using the following method. (3x2)/(x3)(x+1), the roots of the denominator are x = 3 and x = 1. Representing as partial fractions we get the...

Math
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Math
l 2m +3 l <12 We need to find the values of m such that m belongs to Z. By definition we will rewrite: ==> 12 < 2m+3 < 12 We will subtract 3 from both sides. ==> 15 < 2m...

Math
Given the points (x,2) and (2,4). The distance between the points is 5 units. We need to find the values of x We will use the distance between two points formula to find the values of x. We...

Math
Given the graph x^2  5x  1 and the line y= 2x7 We need to find the intersection points between the curve and the line. The points of intersections are the points that verifies both...

Math
Given the quadratic function 3x^2 5x 7 = 0 We will use the quadratic formula to find the roots. We know that the quadratic formula is given by: ==> x = [ b + sqrt(b^24ac)]/2a a = 3 b= 5...

Math
To find which value if greater, we will calculate the value of each percentage. ==> 10% of 325 = 10/100 * 325 = 32.5 ==> 15% of 295 = 15/100 * 295 = 44.25 ==> 25% of 250 = 25/100 * 250 =...

Math
Given the curve x^2 5x +1 We need to find the area bounded by the curve and the lines x= 1 and x= 2 Then we will find the integral of the curve between 1 and 2 Let F(x) = Int (x^2  5x +1) Then...

Math
Given that g(x) = (2x1)/(3x+2) We need to find the f'(0) First we will find the first derivative f'(x) ==> Let f(x) = u/v such that: u= 2x1 ==> u' = 2 v= 3x+2==> v' = 3 ==> f'(x)...

Math
Given the curve f(x) = 3x^2 + 12x 1 We need to find the extreme value. First we will look at the coefficient of x^2 and we note that the sign is negative, then the function has a maximum...

Math
Given that the sum of a and b is 35 Then, we will write: a + b= 35.........(1) Now we know that b is one more than a. ==> b = a+ 1......(2) Now we will substitute (2) into (1). ==> a + (a+1)...

Math
3x4y = 5.........(1) 2x+y = 7 ...........(2) We will use the elimination method to solve for x and y. First we will multiply (2) by 4 and add to (1). ==> 4*(2) 8x + 4y = 28 Now we will add to...

Math
The height and the radius forms a right angle triangle such that the hypotenuse is h and the sides are 11 and 16. We will calculate the length of the hypotenuse. ==> h= sqrt( 11^2 + 16^2) ==>...

Math
You need to use the information provided by the problem, such that: `{((fof)(x) = (x1)+1/(x^2+x+1)),((fof)(x) = 0):} => (x1)+1/(x^2+x+1) = 0` You need to bring the terms in equation to a...

Math
You need to evaluate the indefinite integral such that: `int 1/(root(3) (x  2))dx = int (x  2)^(1/3) dx` `int (x  2)^(1/3) dx = ((x  2)^(1  1/3))/(1  1/3) + c` `int (x  2)^(1/3) dx =...

Math
You also may use elimination method, hence, if you need to eliminate the variable x, you need to multiply by 2 the first equation and add the new equation to the second equation of the system,...

Math
You need to substitute `x + 2` and ` x  2` for x in equation of function `f(x) = mx + n` , such that: `f(x + 2)*f(x  2) = (m(x + 2) + n)(m(x  2) + n)` `f(x + 2)*f(x  2) = (mx + n + 2m)(mx + n...

Math
You need to find the new composed function, hence, you need to replace the equation of v(t) for t in equation of u(t), such that: `(uov)(t) = 2(t  1)^2  1 => (uov)(t) = 2(t^2  2t + 1)  1`...

Math
You should come up with the following substitution, such that: `e^x  1 = u^2 => e^x dx = 2udu => dx = (2udu)/(1 + u^2) ` Changing the variable yields: `int (e^x  1)^(1/2)dx = int...