
Math
In the standard form (3i)/(2+7i) would have a real denominator. (3i)/(2+7i) => (3i)(2  7i)/(2+7i)(2  7i) => [6  2i  21i + 7i^2)/(4  49i^2) => (23i  1)/53 => 1 / 53 ...

Math
The standard equation of a circle is x^2 + ax + y^2 + bx + c = 0 As it passes through (8, 0), (0, 6) and (0, 0) substituting the coordinates of the points in the equation we get 3 equations c = 0...

Math
The inequality to be solved is : x^3 < x x^3 < x move all the terms to the left => x^3  x < 0 factor the left hand side => x(x^2  1)< 0 => x(x  1)(x + 1) < 0 Now the left...

Math
You need to express `cos A = cos 2(A/2), ` hence `cos A = cos^2 (A/2)  sin^2 (A/2)` `` `sin A = 2sin(A/2)cos(A/2)` `` You need to prove that `tan (A/2) = (1 + cos A)/sinA` `tan(A/2) = (1 + cos^2...

Math
We have to solve the system of equations: x + y = z ...(1) x  2 = y ...(2) x^2 + y^2 = 8 ...(3) From (2) we get y = x  2 substitute in (3) x^2 + (x  2)^2 = 8 => x^2 + x^2 + 4  4x = 8 =>...

Math
We have to solve x + y = 1 and x^2 + 4y = 18 x + y = 1 => x = 1  y substitute in x^2 + 4y = 18 => (1  y)^2 + 4y = 18 => 1 + y^2  2y + 4y = 18 => y^2 + 2y  17 = 0 y1 = 2/2 + sqrt (4...

Math
The system of equations to be solved is 4x + 6y + z = 0 ...(1) x + 2y + z = 0 ...(2) x + 3y + 4z = 2 ...(3) (3)  (2) => y + 3z = 2 (1)  4*(2) => 4x + 6y + z  4x  8y  4z = 0 => 2y ...

Math
The set of simultaneous equations to be solved is: x + y + z = 0 ...(1) 2x + 2y + 2z = 0 ...(2) x + 3y + 4z = 2 ...(3) It can be seen that dividing all the terms of (2) 2x + 2y + 2z = 0 => x +...

Math
The sum of the three angles of all triangles is equal to 180 degrees. Let A, B and C denote the angles of the triangle ABC. B is 4 times A => B = 4A C is 13 more than 5 times A => C = 5A +...

Math
We have to solve 4^x = 8 for x. Using logarithms this can be done in the following way. 4^x = 8 take the log of both the sides => log (4^x) = log 8 use the property of log that log a^b = b*log...

Math
Here we have log(x) 2 = 0.5, where the base of the log is x. log(x) 2 = 0.5 use the relation that log(a) b = c => b = a^c => 2 = x^(1/2) => 2^2 = x => x = 4 The solution is x = 4.

Math
We have to solve log(3) x = 5 , where 3 is the base of the logarithm as you have stated. log(3) x = 5 use the relation that log(a) b = c => b = a^c => x = 3^5 => x = 243 The solution is x...

Math
We have to solve 7.3(2.7)^x+2 = 60.225 for x. 7.3(2.7)^x+2 = 60.225 => 7.3(2.7)^x = 60.225  2 => (2.7)^x = (60.225  2)/7.3 take the log of both the side and use the property of logs: log...

Math
A quadratic equation is of the form ax^2 + bx + c = 0, where the coefficients a, b and c are real. A quadratic equation may take complex values for x but the coefficients are always real. Now...

Math
The equation to be solved is: 1/2 + x/6 = 18/x 1/2 + x/6 = 18/x multiply all the terms with x => x/2 + x^2/6 = 18 take all the terms to one side and eliminate the denominators => 3x + x^2 ...

Math
At the point of intersection the x and y coordinates are the same. To find the point of intersection we solve y=5x and y=(13x^2)^1/2 substitute y = 5  x in y=(13x^2)^1/2 => 5  x = (13 ...

Math
We require the result of (75i)(7+8i). Open the brackets and multiply (75i)(7+8i) => 49 + 35i + 56i  40i^2 use i^2 = 1 => 49 + 35i + 56i + 40 => 9 + 91i The product is 9 + 91i

Math
We need to find n if (n+2)(n3)=5/2 (n+2)(n3)=5/2 => n^2 + 2n  3n  6 = 5/2 => n^2  n  6 = 5/2 => 2n^2  2n  12 = 5 => 2n^2  2n  17 = 0 n1 = 2/4 + sqrt(4 + 136)/4 => 1/2 +...

Math
The discriminant of an equation ax^2 + bx + c = 0 is b^2  4ac Here we have x^2+11x+121=x+96 => x^2 + 10x +25 = 0 b^2  4ac = 100  100 = 0 The discriminant is 0.

Math
The hypotenuse of the right triangle is 10 units and the angle B = 70 degrees. Let the angle which equals 90 degree be C. The length of the side b is sin 70 = b/ 10 => b = 9.396 units The angle...

Math
Given the function : y= 6/sqrt u= We will use the quotient rule to find the derivative. Let y= t/v such that ==> t= 6 ==> t' = 0 ==> v = sqrtu ==> v' = 1/2sqrt(u) ==> y' = ( t'*v ...

Math
Let L be the length and W be the width. We can say that L = W + 2 And we can say that L*W = 80 Now we can substitute the value of L from the first equation into the second. W*(W+2) = 80 That gives...

Math
log8 (y+2) = 3 First we will determine the domain. ==> y+ 2 > 0 ==> y > 2 ...........(1) First we will rewrite using the exponent form. ==> y+ 2= 8^3 ==> y+ 2 = 512 Now we will...

Math
We have to prove : sin^4x  cos^4x = 2sin^2x  1 Let's start from the left (sin x)^4  (cos x)^4 => (sin x)^4  [ 1  (sin x)^2]^2 => (sin x)^4  1  (sin x)^4 + 2*(sin x)^2 =>  1 +...

Math
We have to solve: log(6) ( x+41 )  log(6) (x+1) = 2 use the property log a + log b = log(a*b) log(6) ( x+41 )  log(6) (x+1) = 2 => log(6)[( x + 41)/(x + 1)] = 2 => (x + 41) / (x + 1) = 6^2...

Math
We have to prove that tanx = 2sinx*cosx/(1+cos^2xsin^2x) Let's start from the right. 2sinx*cosx/(1+cos^2xsin^2x) => 2*sin x * cos x /( 1 + (cos x)^2  (sin x)^2) => 2*sin x * cos x /( 1 +...

Math
You have not provided a proper question. It is given that sin x*cos 150 = 0 but 5  cos x*sin 150 does not have any purpose. sin x*cos 150 = 0 can be solved for x as cos 150 is not equal to 0, only...

Math
If you use FOIL for (x2)(5x+2) you get 5x^2  10x + 2x  4 = 5x^2  8x  4 For (3x+5)(x2)+(2x3)(2x2) you get 3x^2 + 5x  6x  10 + 4x^2  6x  4x + 6 simplifying we get 7x^2  11x  4 This is...

Math
We need to prove that : cosx/ (1sinc)  sec x = tanx We will start from the left side and prove the right side. First, we know that sec x = 1/cosx ==> cosx / (1sinx)  1/cosx ==> Now we...

Math
It is given that 2*sin3A = sin18/sec72 + sin72/sec18 And you surely mean 0 < A < pi/4 as theta has not been used anywhere. => 2*sin3A = sin18*cos 72 + sin 72*cos 18 => 2*sin3A = sin (18...

Math
If FOIL method is used for (x  2)(5x + 2) the result is 5x^2 + 2x  10x  4 => 5x^2  8x  4 ...(1) If FOIL method is used for (3x + 5)(x  2) + (2x  3)(x  2) we get 3x^2  6x + 5x  10 +...

Math
We'll rewrite the equation to create common factors. Foir this reason, we'll replace the term 7x^2 by the difference 2x^2 – 9x^2. x^4  2x^3 + 2x^2 – 9x^2 + 18x  18 = 0 We'll group the...

Math
We'll rewrite the middle terms as 17x = 25x – 8x. We'll rewrite the equivalent equation:5x^2 – 8x + 25x – 40 = 0 We'll create pairs of terms:(5x^2 – 8x) + (25x – 40) = 0 We'll factorize...

Math
A quadratic equation can also have complex roots. In that case too it would not be possible to arrive at the solution by factoring; instead we would have to use the quadratic formula. Complex roots...

Math
The slope and yintercept form can be found by writing the equation as a form where y is expressed in terms of x and is the dependent term and x is the independent term. The format is the...

Math
To compute the expression, we need the values of coefficients of the quadratic a,b,c. The general form of the quadratic equation is ax^2 +bx + c = 0. Comparing the given equation with the general...

Math
For a circle with radius r, the circumference of the circle is 2*pi*r and the area is pi*r^2. The circle we have here has an area of 36*pi, this gives the area of the circle as 6. The circumference...

Math
The series given is 4 , 44 , 444 , 4444 ,… This series is none of the general ones that we encounter like arithmetic series, geometric series or harmonic series. To find the nth term look at each...

Math
The function f(x) = (7x5)/(x+5) For those values of x which make the first derivative of a function positive it is increasing and when the first derivative is negative it is decreasing. f’(x) =...

Math
The identity to be proven is (cos x + sin x) / (cos x  sin x) = tan [x+(pi/4)] Start from the right hand side tan [x+(pi/4)] expand tan (x + pi/4) => (tan x + tan pi/4) / (1  tan x* tan pi/4)...

Math
The value of lim x>a [ (a^2 – x^2)/(sqrt x – sqrt a)] has to be determined. Subsituting x = a, gives us 0/0 which cannot be determined. Notice that x^2 – a^2 = (x – a)(x + a) = (sqrt x...

Math
To find the median we need to arrange the given data points in ascending or descending order. The data points we have are 101, 102, 100, 100, 110, 109, 109, 108, 109 Arranging them in ascending...

Math
For any equation complex roots are always found in conjugate pairs. If 2+ i and 2 – 3i are roots of the equation so are 2 – i and 2 + 3i This gives the equation as: (x – (2 – i))(x – (2 +...

Math
The function of which we need to find the integral is (3x^2 – 18x + 26)/(x4)*(x3)*(x2) This can be done using partial fractions, but it is a pretty long method. Multiplying the terms of the...

Math
The first agency has a flat charge of $25 and 13 cents per mile. The second only has a per mile charge which is 17 cents. Let the distance that the car has to be driven for to make the amount...

Math
The line given has a slope 3 and the xintercept is 6. This means the line passes through (6 , 0) The equation of the line is y / (x – 6) = 3 => y = 3x – 18 As the line passes through (1 ,...

Math
The series given is 10 , 27, 52, 85,… Subtracting subsequent terms gives 27 – 10 = 17 52 – 27 = 25 85 – 52 = 33 The difference we get does not provide any information about the series....

Math
A point with the variable coordinates (x, y) which is at a constant distance from a fixed point has a locus which is a circle. In the problem the point (x, y) has a constant distance from the the...

Math
Given that the area of the fenced rectangular field is 338 ft^2. Let is assume that the length is L and the width is w. ==> L*W = 338 ..........(1) But we know that L = 2w ==> Then we will...

Math
log (x+5) = 2 log (2x). I believe that you need to determine the values of x that satisfies the equation. We will use logarithm properties to solve for x. First we will add log 2x to both sides....