# Math

• Math
Given the expression E = 200m^4 + 80m^3 + 8m^2. We need to simplify as mush as possible. First we will need to factor all terms is order to find a common factor. Let us factor each term. 200m^4 =...

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Let the integers be x, y and z. Given that the sum of the three integers is 66. ==> x + y + z = 66..............(1) Also, given that the second integer (y) is 2 more than the first (x). ==> y...

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Using the properties of absolute value yields: Hence, evaluating the solutions to absolute value equation yields and .

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Given y= (x-2) / (x^2 - 3) We need to find y'. Since y is a quotient, then we will use the quotient rule to find the derivative. ==> Let y= u/v such that: u= x-2 ==> u' = 1 v= x^2 -3...

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You need to evaluate the determinant of the matrix to check if there exists a inverse for the given matrix, such that: det A = [(1,2,-1),(3,7,-1),(-5,-7,-15)] = -26 != 0 Since the determinant of...

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The right triangle has two sides of length 4 and 3. The third side can have a length which can be calculated using the Pythagorean Theorem. As it is not given that 4 and 3 are the lengths of the...

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The sum of 3/4 + 3/20 can be calculated in the following way 3/4 + 3/20 make the denominator the same, here it would be 20 3*5 / 4*5 + 3/ 20 => 15/20 + 3/ 20 => (15 + 3) / 20 => 18/20...

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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 =...

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In a decimal number each digit is a power of 10, for example in 123 is equal to 1*10^0 + 2*10^1 + 3*10^2. For binary numbers the base is 2 instead of 10. So 11001011 = 1*2^0 + 1*2^1 + 0*2^2 + 1*2^3...

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If an amount P is invested for n terms at a rate of interest of r, it grows to P*(1 + r)^n. Here, the annual rate of interest is 4%, as the compounding is done semi-annually, the effective rate for...

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The area of a triangle if two sides and the enclosed angle are given is (1/2)*s1*s2*cos A Here the sides are 6 yd and 2 yd and the enclosed angle is 10 degrees. The area of the triangle is...

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The equation to be solved is 8^(6x+5)-2/8=0 8^(6x+5) = 2/8 => 2^3^(6x + 5) = 1/4 = 2^-2 As the base 2 is the same, equate the exponent 3*(6x + 5) = -2 18x + 15 = -2 => 18x = -17 => x =...

• Math
Let the width of the rectangle be x, then the length is 3x +2. Now we are given that the area if 33. Then we know that x*(3x+2) = 33 Let us solve. ==> 3x^2 + 2x = 33 ==> 3x^2 +2x - 33 = 0...

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We know that cos (pi/6) = (sqrt 3)/2 cos (pi/6) = cos 2*(pi/12) = 2*[cos (pi/12)]^2 - 1 => (sqrt 3)/2 = 2*[cos (pi/12)]^2 - 1 => 2*[cos (pi/12)]^2 = 1 + (sqrt 3)/2 => [cos (pi/12)]^2 = 1/2...

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We have to solve the equation: x/(x+4) + 4/(x+4) + 2 = 0 x/(x+4) + 4/(x+4) + 2 = 0 => x/(x+4) + 4/(x+4) = -2 => (x + 4)/(x + 4) = -2 => 1 = -2 This is not possible. The given equation has...

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The endpoints of the diameter of the circle are (4, 7) and (7, 11). The distance between the points or the length of the diameter is given as: sqrt[ (4 - 7)^2 + (7 - 11)^2] => sqrt[3^2 + 4^2]...

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We have to find the value of Sum(100 + k), for k = 1 to 100 Sum(100 + k), k = 1 to 100 => 100*100 + Sum(k), k = 1 to 100 The sum of the numbers from 1 to n is given by n*(n + 1)/2 => 100*100...

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First (A + B + C)/3 = 2709 so (1345 + 568 + x)/3 = 2709 1345 + 568 + x = 3*2709 1913 + x = 8127 x = 8127 - 1913 x = 6214 Check - (1345 + 568 + 6214)/3 = 8127/3 = 2709 So the answer is 6214.

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The answer is if a>b>0 then a^n > b^n Proof by induction a) a>b ==> a^1 > b^1 by definiton.b) suppose a^(n-1) > b^(n-1)c) We know that if a>b>0 and c>d>0 then...

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Given the lines: ax + by + c = 0 We need to determine the x and y intercepts and the gradient. The x-intercept is when the line meets the x-axis. Then, y= 0 ==> ax + c = 0 ==> x = -c/a Then...

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1. Simplify (a^2b^0c)^0 Anything to raised to the zero is equal to one, so the answer here is 1. 2. Simplify x^{-2}y^3 We can write x^{-2} as \frac{1}{x^2} ,so this simplifies to...

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Zeller's algorithm is  h=(q+|__((m+1)26)/10__|+Y+|__Y/4__|+|__C/4__|-2C) mod 7 Where q is the date of the month m is the month (March=3, April=4, ... January=13, February=14) Y is the year of the...

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2(X-Y)^2 - 9(X-Y)Z - 5Z^2 = 0 Multiply by -1 both sides. ==> 5z^2 +9(x-y)z -2(x-y)^2 = 0 Let (x-y)= A ==> 5z^2 +9Az- 2A^2 = 0 ==> (5z +A)(z-2A) = 0 ==> (5z+(x-y))(z-2(x-y)) =...

• Math
f(x)=2(x^2-1)^3 f'(x)=2*3(x^2-1)^2(2x)=12x(x^2-1)^2 f''(x)=12(x^2-1)^2+12x(2)(x^2-1)(2x) =12(x^2-1)^2+48x^2(x^2-1) =60x^4-72x^2+12 As a check, expand the binomial and then take the...

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a_1 = -3  a_(k+1)=a_k^2 a_2 = a_1^2 = (-3)^2 = 9 a_3 = a_2^2 = (9)^2 = 81 a_4 = a_3^2 = (81)^2 = 6561 a_5 = a_4^2 = (6561)^2 = 43046721 So the answer is -3, 8, 81, 6561, 43046721

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f(x)= x^3 -5x^2 +6x +3 We know that the slope of the tangent line is the derivative at the point of tendency. But we know that the tangent line is horizontal. Then, the slope is 0. ==>...

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The vertex of the quadratic function f(x)=ax^2+bx+c is (-1, 7) and it passes through the point (-3, -9). The function can be written as f(x) = a(x + 1)^2 + 7 => a(x^2 + 2x + 1) + 7 => ax^2 +...

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3x^4 +2x^3 -9x^2 -12x -4 = 0 We will factor x^3 from the first 2 terms and -1 from the last three terms. x^3(3x+2) - (9x^2 +12x +4) = 0 Now we know that (9x^2 + 12x +4) is a complete...

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When 5000 is loaned at an interest rate of 10% per annum compounded annually, the principal increases every year by an amount equal to the interest earned on the principal in the last period. The...

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We have to determine 75/61 + 42/18 75/61 + 42/18 First equate the denominator of both the terms (75*18)/(61*18) + (42*61)/(61*18) => 1350/(61*18) + 2562/(61*18) => 3912/(61*18) =>...

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Find the slope of the line through the points (-2,5) and (1,4). m=(y_2-y_1)/(x_2-x_1)=(5-4)/(-2-1)=-1/3

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I will assume that the base of the tissue is a square with the length of the side is 8.5 and the height is 7.7 First we will find the volume: ==> V = L X W X H = 8.5 X 8.3 X 7.7 = 556.325 cubic...

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Find the vertical asymptotes for f(x)=(x+8)/(x^2-100) . Note that vertical asymptotes occur when a function with no common factors in the numerator and denominator has a factor that causes a...

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6x^5 -51x^3 -27x  First we will factor 3x . ==gt 3x( 2x^4 -17x^2 -9)  Now we will factor the quadratic equation. ==gt 3x(2x^2 +1)(x^2-9)  ==gt 3x(2x^2+1)(x-3)(x+3)

• Math
Given the expression: 27x^3 + 216 We need to solve for x. ==> 27x^3 + 216 = 0 We know that: (a^3 + b^3)= (a+b)(a^2- ab + b^2) (27x^3 + 216)= ((3x)^3 + 6^3)  ==gt a = 3x ==gt b= 6  ==gt...

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(y^3-64)/(3y^2 -17y +20) First we will factor the numerator and denominator. ==> We know that (a^3-b^3)= (a-b)(a^2+ab+b^2)  ==gt (y^3-64)= (y-4)(y^2 +4y +16)  ==> Now we will factor...

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3a(x+5)= 114 given that a = 4 ==> 3*4 (x+5)= 114 ==> 12(x+5)= 114 Now we will divide by 12. ==> (x+5)= 114/12 = 19/2= 9.5 ==> x = 9.5 -5= 4.5 ==> x = 4.5

• Math
-(3+i) + (7-3i)= We will expand brackets. ==> -3 -i + 7 - 3i = Now we will combine like terms. ==> (-3+7) + (-i-3i) = ==> 4 + (-4i) Then, the final form of the complex number is as...

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If you have 11 counters you can only form 1 array: 1x11. With ten counters, you can form a 1x10 array, as well as a 2x5 array. Thus even though 11>10, you can form more arrays with ten counters...

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The figure shows that the height of the cylinder is equal to the diameter of the sphere. The volume of the cylinder is V = Area of the base*height The base of cylinder is a circle. The radius of...

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You need to find the coefficient n if f = (x+1)g. Replacing f and g by the corresponding equations yields: x^3-nx+1 = (x+1)(x^2-x+1) You need to open the brackets to the right side: x^3- n x+1 =...

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A tribonacci sequence can be defined as t_n=t_(n-1)+t_(n-2)+t_(n-3) for n>4 or as a,b,c,a+b+c,a+2b+2c,2a+3b+4c,... Here we have 5,11,_,_,_,_,_,260,... Let the third element be x; then the...

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You need to solve the integral int (xdx)/(x+1) Adding and subtracting 1 to numerator yields: int ((x + 1 - 1)dx)/(x+1) = int ((x+1)dx)/(x+1) -int dx/(x+1) int (xdx)/(x+1) =int dx - ln |x+1| +...

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You need to know that the rational number pi expresses the relation between the circumference and the diameter of a circle. I suggest you to try the following experiment. Calculate the...

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You should come up with the following substitution such that: e^x = t =gt e^x dx = dt . If you change the variable, you should calculate the limits of integration. x = 0 =gt e^0 = 1 = t  x =...

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The given equation is: Ax = C-By ==> By = -Ax + C In order to rewrite in the slope form, we need to isolate y on one side and determine the coefficient of x which is the slope. ==> To isolate...

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d/dx log x^2 Chain rule: d(inside)*d(outside with inside unchanged) 2x*1/(x^2ln10) =2/(xln10) ----------- Alternatively you could have used a log rule to pop the 2 out front in the...

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It is given that: 2x^3+3x^3y^2 = 4y^2+1. To determine (dy)/(dx) use implicit differentiation. (2x^3+3x^3y^2)' = (4y^2+1)' => 6x^2 + 3x^3*2y*(dy)/(dx) + 9x^2*y^2 = 8y*(dy)/(dx) =>...

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7x^3 + 2x^2 +28x + 8 = 0  First we will re-arrange terms. (7x^3 + 28x) + (2x^2 + 8) = 0  Now we will facotor each terms. Factor 7x from the first two terms and 2 from the last two terms....

• Math
The solution of the given linear equations is: x/14 - 1/2 = -1/7 => x/14 = -1/7 + 1/2 => x/14 = (7-2)/14 => x = 5 (4/5)x + 3 = 2x - 3/5 => 2x - (4/5)x = 3 + 3/5 => `1.2x...