
Math
We have the sum of n terms of a string given as 5n^2+6n Now 5n^2+6n = 5n^2 + 5n + n => 5n(n+1) + n => 10n(n+1)/2 +n Now 10*n(+1)/2 is the sum of n terms of the form n multiplied by 10 and n...

Math
Let E = (x^26x+5) / (x+3) * (x^29)/(x5) First we will multiply numerators and denominators: ==> E = (x^26x+5)*(x^29)/(x+3)(x5) Now we will factor the numerator: ==> E =...

Math
x^2 + 13 = 0 We will factor using the difference between the squares. We know that: a^2 b^2 = (ab)(a+b) let us rewrite: x^2  (13) = 0 Now we will factor: ==> (x^2(13) = 0 ==>...

Math
We need to find the area between the curve y = x^2 + 3 and the tangent line at x= 2. ==> y(2) = 2^2 + 3 = 7 Let us find the tangent. ==> y' = 2x ==> y' = 2*2= 4 Then the equation of the...

Math
We'll subtract g(x) both sides of the inequality that has to be demonstrated. f(x)  g(x) >= 0 We'll substitute f(x) and g(x) by their expressions: x*arctanx  ln(1+x^2) >= 0 We'll assign a...

Math
Let the two numbers be A and B. Now it is given that their GM is 4. This gives us sqrt (A*B) = 4 => AB = 4^2 = 16 We also know that their HM is 16/5, so 2*AB / (A+B) = 16/5 => 2*AB / (A+B) =...

Math
We have the three lines which form the sides of the triangle as 11x7y=81...(1) 3x5y=15...(2) x+4y=12...(3) We have to find their points of intersection. 11*(2)  3*(1) => 33x  55y  33x +...

Math
To find the extreme value of a function f(x) we need to find the first derivative of the function and equate it to zero. Then this is used to solve for x. Here, f(x)=x^4 + 8x^2  48x + 19 f'(x) =...

Math
We have the two functions f(x)=2^x and g(x)=log x. The combined function y = f(x)*g(x). y = 2^x * log x Now the range of a function is the set of all values of y that can be obtained by using the...

Math
We answer the given question we use the formulas: 2 sin A sin B =  cos (A + B) + cos (A  B) and 2 cos A sin B = sin (A + B)  sin (A  B) Now 4*sin ((A+B)/2)*sin ((B+C)/2)*sin ((C+A)/2) =>...

Math
You have not checked the question before submitting it. I think it should be "How much is 6y if y = 3?" and made the appropriate change. The answer to that question is: 6y is equal to six times y....

Math
I really cannot comment on how the earlier teacher explained things. Instead, I'll try to explain the concept of a sphere starting from a circle. The equation of a circle drawn in the xy plane is...

Math
We have to find the sum of k(k+3) with k = 1 to k = n. Now Sum[ k(k+3)] => Sum [ k^2 + 3k] => Sum [k^2] + 3*Sum [k] For k equal to 1 to n => n(n+1)(2n+1)/6 + 3*n*(n+1)/2 =>...

Math
We have to find the remainder when f=3x^42x^3+x^2+ax1 is divided by (x1)^2 (x1)^2 => x^2  2x + 1 We use long division to divide 3x^42x^3+x^2+ax1 by x^2  2x + 1. x^2  2x + 1 ...

Math
Int f(x)dx (x=1 to x=1) = Int [f(x) + f(x)]dx (x = 0 to x = 1). We'll calculate: f(x) + f(x) = 1/(e^x+1)(x^2+1) + 1/(e^x+1)(x^2+1) f(x) + f(x) = [1/(x^2+1)][1/(e^x+1) + 1/(e^x+1)] f(x) +...

Math
We have to simplify : 3a  [(4  3a)/5]  [(a  4)/6] 3a  [(4  3a)/5]  [(a  4)/6] Open the brackets => 3a  [4 /5  3a/5]  [a/6  4/6] => 3a  4/5 + 3a/5  a/6 + 4/6 add terms with a and...

Math
The function is continuous over the interval [3;4] (the function is discontinuously for x = 1 and x = 2). Also, the function, being continuous, it could be differentiated over the range [3 ; 4]....

Math
The asymptote is that line that the curve approaches, both never intersecting each other. Since the asymptote is a line, it's equation is: y = mx + n m = lim f(x)/x, x> +infinite If m exists,...

Math
The roots of the equation x^2  x  a = 0 are x1 = 1 / 2 + sqrt ( 1 + 4a)/2 x2 = 1/2  sqrt ( 1+ 4a) / 2 Also x1^4 + x2^4 = 1 => (1 / 2 + sqrt ( 1 + 4a)/2)^4 + (1 / 2 + sqrt ( 1 + 4a)/2)^4 = 1...

Math
We have the functions f(x)=x+1 and g(x)=1x. Now f(x) / g(x) > 0 => (x +1)/(1x) >0 This is true if either both x +1 and 1x are greater than 0 or both are less than 0. Taking the first...

Math
We have to findx given that fof(x)=0 and f(x)=1/(x^2+x). fof(x) = f(f(x)) => f(1 / (x^2 + x)) => 1/((1/ x^2 + x)^2 + (1/ x^2 + x)) => (x^2 + x)^2 / (x^2 + x + 1) Now this is equal to zero...

Math
We have to solve x+y+2xy=11 ...(1) 2x^2y+2xy^2=12 ...(2) (2) => x^2y + xy^2 = 6 => xy(x+y) = 6 (1) => x+ y + 2xy = 11 If we take the variables x+y and xy together as A and B, we get...

Math
The equation (xg)^2+(yh)^2 +(zk)^2 < = r^2 all the set of points on and inside a sphere with centre (g,h,k) and radius r. So (x1)^2+y^2+(z3)^2 < = 9 = 3^2 represents a sphere with...

Math
Given the lines: y + x 2 = 0 6y3x + 8 =0 We need to determine the relation between the lines ( parallel, perpendicular, or neither) First we will use the slope to find the relation. If the slopes...

Math
Given the terms: (x2) , 6, (2x3) are terms of an arithmetical progression. Then, we will assume that "r" is the common difference between terms. ==> 6 = (x2) + r ==> x+r =...

Math
Given the inequality: u^2  2u + 1 < 16 First we will factor the quadratic equation. We know that (u1)^2 = u62  2u + 1 => (u1)^2 < 16 Now we will take the square root for both sides....

Math
Given the the right angle triangle ABC such that AC is the hypotenuse. Then, the right angle is B. ==> Given the legs are: AB = 6 BC = 8 Then, we will calculate the hypotenuse AC using the...

Math
Given the quadratic equation f(x) = x^2 + kx 5 We need to find the values of k where the function has 2 real roots. We know that the function has 2 real roots when delta > 0 delta = b^2  4ac...

Math
Given the line y = ax8 Also, given that the line (y) intercepts with the xaxis at the point x=2. Then the values of y when x =2 is 0. Then, we know that the point (2,0) is on the line y. Let us...

Math
Given the derivative f'(x) = (x^3 2) /x^4 We need to find f(x). We know that the integral of f'(x) = f(x). ==> f(x) = intg (x^32)/x^4 dx Let us simplify f'(x). ==> f(x) = intg (x^3/x^4) ...

Math
Given the function g(x) = 6x^3  4x^2 + 8x 3 We need to find the second derivative g''(x). First we will differentitae g(x). ==> g'(x) = (6x^3)'  (4x^2)' + (8x)' (3)' =...

Math
Given the function f(x) = x^2 + 2x 1 and the line y= 2x+3 We need to find the intersection points for the curve and the line. Then, we need to find the point that verifies f(x) and y. ==> f(x)...

Math
Given the equation: 8*2^3x = 4^(x1) We need to find x value. First we need to simplify the bases. We know that 8 = 2^3 and 4 = 2^2 ==> (2^3)*(2^3x) = 2^2^(x1) Now we know that x^a * x^b =...

Math
We have the highest point that a person in the Ferris wheel reaches as 43 feet. The diameter of the Ferris wheel is 40 feet. And the time taken to make a revolution is 8 seconds. It is also given...

Math
Let's analyse an example: x^5 The superscript 5 represents the exponent. This one is presenting us how many times the variable x is multiplicated by itself. x^5 =x*x*x*x*x (x multilpied by x...

Math
You may also consider the terms `x^2  2x` where you can factor out `x` , such that: `x(x + 2)` Substituting back in expression `x(x + 2)` for `x^2  2x` yields: `x(x + 2)  x(x + 2)`...

Math
You should follow the order of operations, hence, you need to perform multiplications first and then you need to perform additions and subtractions such that: You need to group the terms that...

Math
P(a,b) = P(1.137629, 1.489691) up to nearst 6th decimal. To the nearest 100th value of b we approximate and give the solution P(a, b) = P(1.14, 1.49) Explanation: We consider a function f(x) =...

Math
cos(3pi/4 + x) + sin(2pi/4  x) = 0 We will use trigonometric identities to solve. We know that: cos(x+y) = cosx*cosy  sinx*siny ==> cos(3pi/4+ x) = cos3pi/4*cosx  sin3pi/4*sinx...

Math
sinx  tany cosx = sin(xy) / cosy We will start from the right side and prove the left side. We will use trigonometric identities to solve. We know that: sin(xy) = sinx*cosy  cosx*siny ==>...

Math
cos(x+y)*cosy + sin(x+y)*siny = cosx First we will use trigonometric identities. We know that: cos(x+y) = cosx*cosy  sinx*siny sin(x+y) = sinx*cosy + sinx*cosy Now we will susbtitute into the...

Math
(tanx + tany)/(cotx + coty) = (tanx)(tany) We will start from the left side. We know that: tanx = sinx/cosx cot(x) = cosx/sin(x) ==> (tanx+tany)/(cot(x)+cot(y)) = (sin(x)/cosx +...

Math
We have to prove that (sec x)^2  (sec y )^2 = (tan x)^2  (tan y)^2. (sec x)^2  (sec y )^2 = (tan x)^2  (tan y)^2 => (sec x)^2  (sec y )^2 = (sin x/cos x)^2  (sin y/cos y)^2 => (sec x)^2...

Math
We have to prove that (cos x)^2 * (cos y)^2 + (sin x)^2 * (sin y) ^2 + (sin x)^2 * (cos y)^2 + (sin y)^2 * (cos x)^2 = 1 Now sin (x+y) = (sin x)*(cos y) + (cos x)*(sin y) and cos (x+ y) = (cos...

Math
csc^2 x + sec^2 x = csc^2 x * sec^2 x We will start from the left side and prove the right side. We know that: csc(x) = 1/sin(x) ==> csc^2 x = 1/sin^2 x sec(x) = 1/cos(x) ==> sec^2 x =...

Math
sin(x)*tan(x) = sec(x)  cos(x) We will start from the left side. ==> We know that tan(x) = sin(x)/cos(x) ==> sin(x)*tan(x) = sin(x)*sin(x)/ cos(x) = sin^2 x/...

Math
Here the equation of the sphere is (x  1)^2 + y^2 + (z  3)^2 <= 9. So a solid sphere has been considered here with the outer surface given by (x  1)^2 + y^2 + (z  3)^2=9. Now a plane y=3...

Math
The shape of the cut off section is circle since it is a crosssection of a sphere. Therefore, the ? is the radius of the cut off section. Now let's figure out "?". If we use Pythagorean theorem,...

Math
We have the function f(x) = (3/2) x – 9/2 and we have to find the inverse function. Let y = f(x) = (3/2) x – 9/2 => y = (3/2) x – 9/2 => 2y = 3x  9 => 2y + 9 = 3x => x = 2y/3 +...

Math
f(x) = (x+1)/(2x3) and g(x) = (x3)/(2x+5) We put g(x) in place of x in (x+1)/(2x3). fog(x) = (g(x)+1)/(2g(x)3) fog(x) = {(x3)/(2x+5) +1}/{{2(x3)/(2x+5)  3}. We multiply both numerator and...