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MathWe'll rewrite the function: f(x) = 1/(x+1)(x+2) The function is continuous over the interval [1;2] (the function is discontinuously for x = 1 and x = 2, that are the roots of the denominator)....

MathWe have to find the sum of the terms An = n^2 + 12n Now Sum [ An], of n terms => Sum [ n^2 + 12n ] , of n terms => Sum [ n^2] + Sum [ 12n] , of n terms => n(n+1)(2n+1)/6 + 12* n(n+1)/2...

MathTo determine the antiderivative, you'll have to evaluate the indefinite integral of the given function y. Int f(x)dx = Int sqrt(25  x^2)dx (y = f(x)) We'll factorize by 25: Int sqrt[25(1 ...

MathGiven the logarithm equation: log(x) 8e^3 = 2 We need to find the values of x. let us rewrite into the exponent form. ==> x^2 = 8e^3 But we know that 8= 2^3 ==> x^2 = (2^3)(e^3) Now we will...

MathGiven that: a^2  b^2 = 8 ............(1) ab = 2..................(2) We need to determine a^4 + b^4 Let us square equation (1). ==> (a^2  b^2)^2 = 8^2 ==> a^4  2a^2 b^2 + b^4 = 64 ==>...

MathFind all zeros of the polynomial P(x) = x^3  3x^2  10x + 24 knowing that x = 2 is a zero of the...Given the polynomial P(x) = x^3  3x^2  10x + 24 We are given that x=2 is one of the root. Then we know that (x2) is one of the factors of P(x). ==> P(x) = (x2) * Q(x) Now we will divide P(x)...

MathGiven the number i^231 Let us simplify and determine the values. First we will rewrite the power. ==> i^(231) = i^(3+228) Now we will use exponent properties to simplify. We know that x^(a+b) =...

MathGiven the quadratic equation f(x) = 2x^2 + 5x k We need to find the value of k such that f(x) has two real solutions. We will use delta to determine the values of k. We know that the function has...

MathGiven the parabola y= x^2  5x + 4 and the line y= 2x2 We need to find the intersection points of the parabola and the line. The intersection points are the values of x and y such that: the...

Matha = ( 3 sqrt2)^1/2 b = ( 3+ sqrt2)^1/2 We need to find the values of b(1/a  b) First we will rewrite: b*(1/a  b) = b/a  b^2 Now we will calculate each terms. ==> b/a = (3+sqrt2)^1/2 /...

MathLet the width of the perimeter be "w" and the length be "L" Given that the perimeter is 160 cm But the perimeter is given by : P = 2L + 2W = 160 We will divide by 2. ==> L + w = 80...

MathGiven that: x^2  y^2 = 10..............(1) x+y = 2..............(2) We need to find the values of x and y. First , we will rewrite equation (1) We know that (x^2  y^2) = (xy)(x+y) ==>...

Math1/xy(x+y+z)+1/yz(x+y+z)+1/xz(x+y+z) The least common denominator is xyz(x+y+z) Since this is not a full equation it does not have "=" sign, we cannot multiply least common denominator. But, we...

MathGiven the equation of the line 3x + 3y = 4 The point ( 2, k) is on the line. Then the point ( 2, k) should verify the equation of the line. We will substitute: 3x + 3y = 4 ==> 3 (2) + 3k = 4...

MathLet us assume that the number is E. ==> E = (11/10) ( 11/11) ( 1 1/12) .... (11/99)( 1 1/100) Let us rewrite the terms by using the common denominator. ==> 1 1/10 = 10/10  1/10 = 9/10...

MathThe property is: log (a) + log (b) = log (ab) The bases of the logarithm must be equal. Let us solve an example. Given the logarithm equation: log (x2) + log x = log 4 We need to solve for x. We...

MathGiven the function f(x) = 3(x10)(x4) We need to find the maximum value. First we will open the brackets and expand the function. ==> f(x) = 3(x^2 14x + 40) = 3x^2 + 42x ...

MathLet us assume that the width of the field is "W" and the length of the field is "L" We are given that the area of the rectangular field is 300 m^2 But we know that the area = L*W ==> L*W = 300...

Mathdifference between solving equation and inequalitydifference between solving equation and inequalityI will explain in an example: We have the equation: 2x + 5 = 9 We will solve the equation by isolating x on the left side. We will subtract 5 from both sides. ==> 2x = 4 ==> x = 2 Then there...

MathLet us assume that the selling Price of the house is X. The real estate agent recieved 6% of the selling price which is 8,880 dollars. Then we will write: 6% (x) = 8,880 ==> (6/100) x = 8880 Let...

MathWe have to simplify 2x/(x^2 6x+9)  1/(x+1)  8/(x^2 2x3) 2x/(x^2 6x+9) => 2x/(x^2 3x  3x+9) => 2x/ [x( x  3)  3(x 3)] => 2x/ (x3)^2 8/ ( x^2  2x  3) => 8/( x^2  3x + x ...

MathGiven the function f(x) = 2ln(2x3) + 2 We need to find the inverse function f^1 (x). Let us assume that f(x) = y ==> y= 2ln(2x3) + 2 The goal is to isolate x on one side . Let us subtract 2...

MathGiven the equation: sqrt(4x)^2 = 5 2x First we will simplify the square root. We know that sqrt x^2 = l x l ==> sqrt (4x)^2 = l 4 x l ==> l 4  x l = 5 2x Then we have two cases: ==>...

MathGiven the equation: f(x) = sqrt(x^2 + 4  4x) We need to rewrite and solve the function using the absolute values. First we notice that (x^2 4x+4) is a complete square and equals (x2)^2 Then we...

MathFind the arithmetic sequence:What is the 32nd term of the arithmetic sequence where a1 = 4 and a6...Let a1, a2, a3, ....., an be terms of arithmetic sequence. Then, we will assume that the common difference is r. Given that a1= 4 and a6 = 29 Then, we know that: an = a1 + (n1)*r ==> a6 = a1...

MathGiven the inequality: l 6 + 4x l =< 7 We need to find the values of x where the inequality holds. We will rewrite the absolute values using the definition: ==> 7 =< (6+4x) =< 7 Now...

MathGiven the expression : E= l 2sqrt3 + 5/2l We need to rewrite the number. Let us calculate the values between the absolute value. ==> E = l ( 4sqrt3 + 5)/2 l = l 4sqrt3 +5 l / 2 ==> We know...

MathGiven the function: f(x) = sqrt(x2) / sqrt(3x) we need to find the domain. Since the function if a quotient, then the denominator can not be zero. Also, the square root must be a values greater...

MathYou need to solve for x the equation `x^4+8x^2=9` , hence, you should complete the square to the left side, using the formula `(a+b)^2 = a^2 + 2ab + b^2,` such that: `x^4 + 8x^2 + 16 = 9 + 16` (add...

MathYou can see the expression as a word in a sentence and the equation as the sentence itself. We can use expressions to form equations. Let's see an example: 2x is an expression. If we plug in values...

Mathx+y=5 xy=3 x^2+y^2=? That means that you will need to us the first two equations to solve for the third. Let us rewrite the third equation. ==> We know that ( x+y)^2 = x^2 + 2xy + y^2 ==> x^2...

MathWe know that a^b = a*a*a*....*a multiplied b times. Now, we'll take decimal logarithms both sides: log a^b = log (a*a*a*....*a) Based on product rule of logarithms, we'll transform the product into...

MathThe term of prime number is given to the number that cannot be divided by other numbers, except 1 and itself. Prime factor is a factor of a number that has the property to be a prime number: Let's...

MathIt is a quadratic and you can solve accordingly. But there is another manner to approach the problem, without using quadratic formula. We'll move the term from the right side, to the left side,...

MathThe slope intercept form of an equation is given as y = mx + b where b is the y intercept and m is the slope. So you already have y = mx + 5 You can find the slope of the line by using two points...

MathWe have to solve 2 cases: 1) 5y + 3 = 8y  8 5y  8y = 3  8 3y = 11 y = 11/3 2) 5y + 3 = 8y + 8 5y + 8y = 3 + 8 13y = 5 y = 5/13

MathThe way of solving the exponential equation depends on the right term b. If b is a multiple of a, then the equation could be solved using the one to one property of exponential function. Let's see...

MathAny spot on a graph is defined by a coordinate with a value on the x axis (horizontal) and the y axis (vertical). An equation like the one you gave can be used to find each point on a given line....

MathYes, it helps! x^9y^9 = (x^3)^3  (y^3)^3 We'll note x^3 = a and y^3 = b As you know, the formula of difference of squares is: a^3  b^3 = (ab)(a^2 + ab + b^2) We'll substitute a and b and we'll...

Mathgeometry problemIn a right triangle, one leg is 4 times the other. The hypotenuse squared is 153....Yes. The area of the triangle is .5(base*height). In a right triangle, the two legs are the base and the height. You can use the Pythagorean Theorem to find the lengths of the legs. Once you...

MathYou should use the formula that factors a difference of two cubes such that: `x^3  y^3 = (x^2 + xy + y^2)(x  y)` Considering `x = root(3)(512x^3) = 8x` and `y = root(3)1 = 1` yields: `512x^3  1...

MathThe consecutive integers are: x,x+1 and x+2 We'll add their squares: x^2 + (x+1)^2 + (x+2)^2 = 110 We'll expand the squares: x^2 + x^2 + 2x + 1 + x^2 + 4x + 4 = 110 We'll combine like terms: 3x^2 +...

MathWe have to solve the following equations by elimination 2x5y+3z=1 ...(1) x+4y2z=9 ...(2) x2y4z=5 ...(3) Let's eliminate x and create 2 equations with y and z. (2)  (3) => 4y  2z + 2y +...

MathThe equation of a line passing through the points (x1, y1) and (x2 , y2) is given by (y  y1) = [(y2  y1)/(x2  x1)]*(x  x1) Here we have x1 = 3, x2 = 1 , y1 = 5 and y2 = 4. Substituting the...

MathWe have to solve the simultaneous equations 7x4y=10 ...(1) 4y=x2...(2) add (2) and (1) => 7x  4y + 4y = 10 + x  2 => 7x = 12 + x => 6x = 12 => x = 2 Substitute this in (2)...

MathWe have to solve the equation: 5(u5)+10=12u+3(u+5) First open the brackets 5(u5)+10=12u+3(u+5) => 5u  25 + 10 = 12u + 3u + 15 bring the terms with u to one side and the numeric terms to the...

MathLet's recall the form of the quadratic equation: ax^2 + bx + c = 0 We'll factorize by a: a(x^2 + bx/a + c/a) = 0 We'll group the first 2 terms and we'll complete the square: a(x^2 + 2*x*b/2a +...

Matha) 000 b)0101 .. hence exactly the same 0's and one's

MathI think by y = 2/3x + 1, you mean y = (2/3)*x + 1. So to find the value of y in this case, multiply x with (2/3) and add 1. Similarly for y = (3/4)x  2, multiply x with (3/4) and subtract 2 from...

MathGiven the equation: 3x+4y = 26...........(1) x  2y = 8 ............(2) First we will multiply (2) by 2 : ==> 2x  4y = 16...........(3) Now we will add (1) and (3) ==> 5x = 10 ==> x =...