
Math
We'll use the binomial raised to cube identity: (a+b)^3 = a^3 + b^3 + 3a*b*(a+b) In this case, a = x1 and b = x2 We'll replace a and b by x1 and x2 and we'll have: (x1+x2)^3 = x1^3 + x2^3 +...

Math
We have to solve: (cos x)^2  (sin x)^2 + sin x = 0 (cos x)^2  (sin x)^2 + sin x = 0 substitute (cos x)^2 = 1  (sin x)^2 => 1  (sin x)^2  (sin x)^2 + sin x = 0 => 1  2*(sin x)^2 + sin x...

Math
Let the terms of a geometric progression be : a1, a2, a3, a4 Given that a3= 1 and the common difference is 1/2. ==> Then we know that: a3 = a1*r^2 1 = a1* (1/2)^2 1= a1/ 4 ==> a1= 4 ==>...

Math
Let us assume that the number is x. Then 3 times square x is 3x^2. And we will add 2 times fifth x ==> 3x^2 + 2x/5 = 3x^2 +(2/5)x ==> All divide by (x+7) ==> [(3x^2 + (2/5)x] / (x+7)...

Math
sqrt(x^2 + 3x 5) = x+ 2 First we will square both sides. ==> x^2 + 3x 5 = (x+2)^2 ==> x^2 + 3x 5 = x^2 + 4x +4 ==> 3x 5 = 4x+4 ==>. x = 9 Let us check. ==> sqrt( 81275) =...

Math
Let the first speed be S1 = 55 mph and the time is T1 = 24 min. The second speed is S2= 135 and the time is T2 = 135 min. We need to find the total distance . Let the total diostance be D = D1 +...

Math
Given that : log a = 12 log b= 3 We need to find the value of the expression : log (a^2*b^3)^6 First we know that log a^b= b*log a. ==> log (a^2*b^3)^6 = 6*log (a^2*b^3) Now we know that log ab...

Math
log3 (12n1)  log3 (2x+7) = 2 First we will find the domain. ==> 12n1 > 0 ==> n> 1/12 ==> 2n+7 > 0 ==> n > 7/2 Then The domain is n > 1/12 .............(1) Now we will...

Math
81^2x = 27^(4x1) First we will simplify by factoring the bases. ==> 81 = 3*3*3*3= 3^4 ==> 27 = 3*3*3 = 3^3 Now we will substitute. ==> 3^4^2x = 3^3^(4x1) But we know that x^a^b= x^ab....

Math
f(x) = (3x^25) / x Let us simplify. ==> f(x) = 3x^2/x  5/x ==> f(x) = 3x  5/x Now we will integrate. ==> F(x) = Int (3x  5/x) dx ==> F(x) = 3x^2/2  5*ln x ==> F(x) = (3/2)x^2 ...

Math
Given the curve f(x) = 2x^2  2 We need to find the area bounded between f(x) and x=1 and x= 2 Then we know that the area under the curve is the integral of f(x) between 1 and 2. ==> Then F(x)=...

Math
Given the polynomial equation: 3u^3  u^2 6u + 2 = 0 First we will factor the equation. We will factor u^2 from the first 2 terms. ==> u^2 ( 3u  1)  6u + 2 = 0 Now we will factor 2 from the...

Math
We have to simplify 256^(log 4 (3)) 256^(log 4 (3)) => 4^4^(log(4) 3) => 4^(4*(log(4) 3)) use the property of logarithm: a*log b = log b^a => 4^(log(4) 3^4) => 4^(log(4) 81) the log of...

Math
The average of the numbers with the incorrect entry is 56. The total of the 20 numbers is 56*20 = 1120. The incorrect value that is entered as 91 is actually 19. The excess that this error adds to...

Math
Let us review the rules for the geometric progression. Let a1, a2, a3, a4, a5, a6 are the first 6 terms is a G.P. Let r be the common difference. Then, we know that> a1= a1 a2= a1*r a3= a1*r^2...

Math
Let the price of the book be B and the price of the DVD be D. Given that 2 books and 3 DVDs costs 91 ==> 2B + 3D = 91 ..............(1) Also, given that the cost is 1 book and 2 DVDs is 52....

Math
Given that 2x, 17, 3x1 are terms is A.P Let us assume that the common difference is r. Then we know that: 17 = 2x + r ...........(1) Also we know that: 3x1 = 17 + r => 18 = 3x +...

Math
Given the curve y= x^2  5x + 3 and the curve y= 2x^2  3x + 4 We need to find the position of the intersection point. First we will determine the coordinates of the intersection points. ==>...

Math
Given the circumference of the pool is 23 feet. Let us calculate the radius. ==> C = 2* r * pi ==> 23 = 2*r * pi ==> r= 23/2pi = 3.66 ft Now we need to determine the circumference of the...

Math
Given the volume of the cylinder is 48 and the height is 7. We will use the volume of a cylinder formula to find the radius of the base. ==> V = r^2 * pi * h ==> 48 = r^2 * pi * 7 ==> r^2...

Math
Given the equation of the circle: x^2 + y^2  8x + 12y = 12 We need to rewrite the equation into the standard form in order to determine the radius. ==> (xa)^2 + (yb)^2 = r^2 Then we will need...

Math
The inequality we have is 54l 2x7l =< 13 5  4l 2x7l =< 13 => 5  13 =< 4*2x  7 => 8 =< 4*2x  7 => 2 =< 2x  7 But we know that x >=0 2x  7 >= 0 This...

Math
y= log3 ( 5x^2125) We need to find the domain. The domain is all x values such that the function y is defined. Since the function is a logarithm, then we know that the logarithm should be greater...

Math
y= (2x3) / (x^2 1) We will use the quotient rule to find the derivative. Let y= u/ v such that" u= 2x3 ==> u' = 2 v = x^2 1 ==> v' = 2x Then we know that: y' = ( u'v uv' ) / v^2 ==>...

Math
An unbiased dice has 6 faces with a different number each of which has an equal probability of showing when the die is tossed. In the problem we have 3 unbiased dice. 1) First we need to find the...

Math
The extreme value of f(x) = 3  4*sin x occurs at the value of x where f'(x) = 0 f'(x) = 0  4*cos x = 0 => cos x = 0 x = arc cos 0 = pi/2 and 3*pi/2 At the maximum point f''(x) is negative...

Math
We have to find the properties of the roots of the given equation without solving it. We can determine if the roots are real or complex and whether they are the same or different. For the equation...

Math
The solution of x/y + y/x = 8 can be obtained by equating z = x/y This makes x/y + y/x = 8 => z + 1/z = 8 => z^2 – 8z + 1 = 0 z1 = 8/2 + sqrt (8^2 – 4) / 2 => 4 + sqrt 60 / 2 => 4 +...

Math
A quadratic equation does not have one root, instead it is referred to as the equation having 2 equal roots. A quadratic equation in the general form ax^2 + bx + c = 0 has the roots given by: x1 =...

Math
The solution of the equation x^3 + 3x^2 + x + 3 = 0 can be found by finding the factors of the polynomial. x^3 + 3x^2 + x + 3 = 0 => x^2(x + 3) + 1(x + 3) = 0 => (x^2 + 1)(x + 3) = 0 x^2 + 1...

Math
To verify sin(x + y) + sin(x  y) = 2 sin x cos y use the formula of the sin of the sum of two angles: sin (x + y) = sin x*cos y + cos x *sin y. sin(x + y) + sin(x  y) => sin x*cos y + cos...

Math
You'll have 4 answers for x in total. (x+10)=1 & (x+11)=1 & (x+12)=1 & (x+13)=1 X=9 & x=10 & x=11 & x=12 Done! :)

Math
Given the quadratic equation : f(x) = 3x^2 + 5x + C We know that f(x) has two complex roots. Then the discriminant is negative. ==> (b^2  4ac < 0 a = 3 b= 5 c = C ==> 25  4*3*C <...

Math
tanx*sinx* (cosx+1) + cosx = sin62 x+ sec x Let us start from the left side and prove the right side. We know that tanx = sinx/cosx ==> sinx/cosx * sinx ( cosx+ 1) + cosx ==> We will rewrite...

Math
Given the equation of the circle is : x^2 + y62  2x + 8y = 13 We need to find the area. First we need to determine the radius. Then, we will rewrite the equation into the standard form. (xa)^2 +...

Math
f(x)= 3x^2  4x + 5 we need to find the extreme value of f(x). First we will find the critical points. ==. f'(x) = 6x 4 = 0 ==> x = 4/6 = 2/3 Then the function f(x) has an extreme value at...

Math
Given y= (2x3) / (x+2) we need to find the intercepts of y. First we will determine the yintercepts. Then x value is 0. ==> y= (03)/(0+2) = 3/2 Then the yintercept is (0, 3/2) now we will...

Math
Given that f(x) = sqrt( 5 l 3x4l ) We need to find the domain of f(x). Since the function is a square root, then the domain is all x values such the function inside the square root is positive or...

Math
One of the roots of f(x) = ax^2  3x + 2 is 3. The function f(x) is equal to 0, when x is equal to any of its roots. The roots of f(x) are b/2a + sqrt (b^2  4ac) / 2a => 3/2a + sqrt (9  8a) /...

Math
There are two integers A and B. A is 3 less than twice B => A = 2*B  3 The sum of the two is 12 => A + B = 12 => 2*B  3 + B = 12 => 3B = 15 => B = 5 A = 12  5 = 7 The integers...

Math
We have : log3(8x+3) = 1+ log3 (x^2) log3(8x+3) = 1+ log3 (x^2) => log3(8x+3)  log3 (x^2) = 1 use the property log a  log b = log (a/b) => log3 [(8x + 3)/(x^2)] = 1 => (8x + 3)/(x^2) =...

Math
Given the derivative f'(x) = (x^3  x)/4x^4 Let us simplify. ==> f'(x) = (x^3/4x^4)  x/4x^4 ==> f'(x) = (1/4x)  1/4x^3 ==> f'(x) =(1/4)[ 1/x  x^3) Now we will integrate. ==> f(x) =...

Math
To solve an equation like a*cos^2x+b*sinx=b, remember the basic identity (sin x)^2 + (cos x)^2 = 1 a*cos^2x+b*sinx=b => a*(1  (sin x)^2) + b*sin x = b => a  a*(sin x)^2 + b*sin x = b =>...

Math
Let the three terms of the AP be a , b and c. Consecutive terms of an AP have a common difference b = a + d and c = b + d = a + 2d If you can determine a and d using the relations for the others in...

Math
Use the formula for the cosine of the sum of two angles cos (a + b) = cos a*cos b  sin a * sin b cos 2x = cos (x + x) => cos x*cos x  sin x * sin x => (cos x)^2  (sin x)^2 This gives cos...

Math
The request of the problem is vague since it is not clear what is needed, hence, supposing that you need to work with a fraction that has a complex number to dennominator, you need to perform first...

Math
The identity that has to be proved is: (tan A)^2/(1+ (tan A)^2) + (cot A)^2/(1+(cot A)^2) = (1 2(sin A)^2 (cos A)^2)/(sin A)(cos A) Starting with the left hand side: (tan A)^2/(1+ (tan A)^2) +...

Math
We need to find x if 5*6^x2*3^2x3*2^2x=0 5*6^x2*3^2x3*2^2x=0 divide all the terms by 2^2x => 5*(3/2)^x  2*(3/2)^2x  3 = 0 let (3/2)^x = y => 5y  2y^2  3 = 0 => 2y^2  5y + 3 = 0...

Math
We have to solve for x and y given that y=6/x and 2^(x+y)=32 substitute y = 6/x in 2^(x+y)=32 => 2^(x + 6/x) = 32 => 2^(x + 6/x) = 2^5 => x + 6/x = 5 => x^2  5x + 6 = 0 => x^2  3x...

Math
The equation to be solved is x=(x+6)^1/2 x=(x+6)^1/2 square both the sides => x^2 = x + 6 => x^2  x  6 = 0 => x^2  3x + 2x  6 = 0 => x(x  3) + 2(x  3) = 0 => (x + 2)(x  3) =...