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

Math
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...

Math
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...

Math
`2(XY)^2  9(XY)Z  5Z^2 = 0` Multiply by 1 both sides. `==> 5z^2 +9(xy)z 2(xy)^2 = 0` Let (xy)= A ==> `5z^2 +9Az 2A^2 = 0` `==> (5z +A)(z2A) = 0` `==> (5z+(xy))(z2(xy)) =...

Math
`f(x)=2(x^21)^3` `f'(x)=2*3(x^21)^2(2x)=12x(x^21)^2` `f''(x)=12(x^21)^2+12x(2)(x^21)(2x)` `=12(x^21)^2+48x^2(x^21)` `=60x^472x^2+12` As a check, expand the binomial and then take the...

Math
`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

Math
`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. `==>...

Math
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 +...

Math
`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...

Math
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...

Math
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)` =>...

Math
Find the slope of the line through the points (2,5) and (1,4). `m=(y_2y_1)/(x_2x_1)=(54)/(21)=1/3`

Math
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...

Math
Find the vertical asymptotes for `f(x)=(x+8)/(x^2100)` . Note that vertical asymptotes occur when a function with no common factors in the numerator and denominator has a factor that causes a...

Math
`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^29) ` `==gt 3x(2x^2+1)(x3)(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...

Math
`(y^364)/(3y^2 17y +20)` First we will factor the numerator and denominator. ==> We know that `(a^3b^3)= (ab)(a^2+ab+b^2) ` `==gt (y^364)= (y4)(y^2 +4y +16)` `` ==> Now we will factor...

Math
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) + (73i)= We will expand brackets. ==> 3 i + 7  3i = Now we will combine like terms. ==> (3+7) + (i3i) = ==> 4 + (4i) Then, the final form of the complex number is as...

Math
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...

Math
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...

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

Math
A tribonacci sequence can be defined as `t_n=t_(n1)+t_(n2)+t_(n3)` 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...

Math
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 +...

Math
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...

Math
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 =...

Math
The given equation is: Ax = CBy ==> 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...

Math
`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...

Math
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)` =>...

Math
`7x^3 + 2x^2 +28x + 8 = 0` `` First we will rearrange 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 = (72)/14` => `x = 5` `(4/5)x + 3 = 2x  3/5` => `2x  (4/5)x = 3 + 3/5` => `1.2x...

Math
`f(x) = 6` is a horizontal line through y = 6. Since it has a slope of zero, the function is not increasing or decreasing. It is a constant function.

Math
Consider the function `f(x)=1/x` . As x tends to 0, the function increases without bound, so Wallis argues that as x crosses the yaxis the function continues increasing; thus these numbers are...

Math
The function `y = x` . For values of `x < 0, y = x` and for values of `x >= 0, y = x` . The graph of the function y = x is: The graph intersects the xaxis at the point (0, 0).

Math
The function f(x) = (x^2 + 6x  9)/(x  6) The vertical asymptotes correspond to the roots of the denominator, x = 6 The function does not have a horizontal asymptote as the degree of the...

Math
You should notice that your problem involves rational exponents, hence you need to remember that the fifth root means one fifth power such that: `root(5)(1215) = (1215)^(1/5)` Notice that the...

Math
Find the sum of a 6term geometric series if `a_1=19` and `a_6=319333` The sum of a finite geometric series is `S_n=a_1((1r^n)/(1r))` for `r!=1` . We have `a_1=19 and n=6` , so we need to find...

Math
200000=100000(1+0.12)^N 2=(1.12)^N Solving for an exponent... time to break out the logs! log 2 = log (1.12^N) = N log 1.12 Did you remember that log rule? `logA^x=xlogA` N = log 2 / log 1.12 =...

Math
Since they need the scenery in half the time, it will require twice as many people. We can set this up to calculate the number of people as well 4 people * 6 days = x people * 3 days x = 8 people

Math
You need to remember that `arcsin(sin x) = x` , hence you may write the equation in terms of x such that: `0.25 = x + x/2 =gt 1/4 = x + x/2` You need to bring the terms to a common denominator such...

Math
Given the points (2,5) and (3,4) First we will find the slope. ==> m= (y2y1)/(x2x1)= (54)/(23)= 1/5 = 1/5 Now we will find the equation of the line . ==> yy1 = m (xx1) ==> y 4 =...

Math
Let the number that 6 is 98% of be X. X*`(98/100)` = 6 => X = `600/98` => X = `300/49` 6 is 98% of 300/49

Math
The local maximum point of the expression x^3  6x^2 + 8 for values of x lying in [2, 2] can be determined by solving (x^3  6x^2 + 8)' = 0. The result x = c must lie in [2, 2] and the value of...

Math
Let the quadratic equation be: `f(x)= ax^2 + bx + c` let x1, and x2 be the roots: Then we know that: `x1+ x2= b/a` `x1*x2= c/a` Given that the roots are: `x1= 5+sqrt2*i ` `x2= 5sqrt2*i` `==gt x1+...

Math
Think of it as `y=x^2+6x^(1)` Then `(dy)/(dx)=2x6/(x^2)` by power rule. Now we must solve `2x=6/x^2` `2x^3=6` `x^3=3` `x=root(3)(3)`

Math
`(^3sqrt(x) sqrt(x^5))/sqrt(25x^16) ` `(x^(1/3)(x^(5/2)))/(5x^8)` `x^(1/3+5/2) / (5x^8)` `==> (x^(17/6))/(5x^8)` `==> (1/5) x^(17/6  8)` `==> (1/5)x^(31/6)` `==> 1/(5x^(31/6))`...

Math
The solution of x^2 + 3x + 8 = 6 has to be determined x^2 + 3x + 8 = 6 => x^2 + 3x + 8 = 6 and x^2 + 3x + 8 = 6 => x^2 + 3x + 2 = 0 and x^2 + 3x + 14 = 0 x^2 + 3x + 2 = 0 => x^2 + 2x...

Math
P= Ae^(rt) Given the initial amount is A= 7500 Also, given the rate r= 4.3% = 0.043 We need to find the time (t) needed in order for the money to double. ==> Then A = 2A = 2*7500 = 15000 ==>...

Math
The expression `sqrt 45 + (20/ sqrt 5)` has to be expressed in the form `k*sqrt 5` , where k is an integer. `sqrt 45 + (20/ sqrt 5)` => `sqrt 45 + (4*5/ sqrt 5)` => `sqrt 45 + (4*sqrt 5)`...

Math
The Edisons are deciding whether or not to sell their 6yearold SUV. Its original price was $39,289 and its annual rate of depreciation is 17%. As the value of the car depreciates by 17% every...