
Math
What you need to do is put this equation that is currently in "general form" (y=ax^2 +bx+c) into what is called "vertex form" (yk=a(xh)^2) To do this, you must follow these steps: 1) Subtract...

Math
Let's call the first number N1 and the second number N2. So, let's call N1 the smaller number and N1=x Let's call N2 the larger number and N2=x+4. X(X+4)=45 x^2+4x=45 x^2+4x45=0 (x+9)(x5)=0...

Math
Triangle ABC is isosceleswith AB= AC. To find BD^2 CD^2. Let D be the point on AC where the circle onAB as diameter intersects. Now AB = AC by data, The angle ADB is a right angle ,as AB is the...

Math
To examine the point x= 1/3 on y(x) = (1/12)(3x+1)^4  8x. At x = 1/3, y(1/3)= (1/12)(3/3+1)^4 8/3= 16/12  8/3 = {328*32}/12 = 4/3. Therefore y(1/3) = 4/3. Therefore y(1/3) = 4/3. Therefore...

Math
If 3x^2, x and 2 are subsequent terms of a GP, there is a common ratio between 3x^2 and x and between x and 2. So we have 3x^2 /x = x /2 => 2*3*x^2 = x^2 When we cancel x^2, we get => 6 = 1,...

Math
We have to find x and y given that (8x + yi) / (1 + 4i) = (2x + i)/ (3 + 2i) (8x + yi) / (1 + 4i) = (2x + i)/ (3 + 2i) => (8x + yi) (3 + 2i) = (2x +i) (1+ 4i) => 24x + 16xi + 3yi + 2yi^2 = 2x...

Math
To find the required area we need to calculate the definite integral of the function y = sqrt x, between the limits x = 0 and x = 25. First let us find the indefinite integral of y = sqrt x y =...

Math
Given the equation:\ 1/8 = 16^x First we will simplify 8 and 16 as powers of the prime number 2. We know that: 8 = 2*2*2 = 2^3 16 = 4*4 = 2*2*2*2 = 2^4 Now we will rewrite into the given equation:...

Math
Let the width of the gravel border be x. Then the rectangle's sides after the gravel is: Length = 10 + x Width = 4 + x The area of the rectangle + the border = (x+10)*(x+4) ==> The area of the...

Math
Given the curve y= 50/(2x1)^2 We need to find the tangent line at the point (3,2) We know that the equation of the tangent line is: (yy1) = m(xx1) where (x1,y1) is any point on the line and m is...

Math
We are given that f(x) = 3x^2  2x + 5 and g(x) = 3x  1. We have to find g(4)  f(2). g(4) = 3*4  1 = 12  1= 11 f(2) = 3*(2)^(2)  2*(2) + 5 = 3*4 + 4 + 5 = 12 + 4 + 5 = 21 Therefore g(4) ...

Math
We have to solve the equation 9^(x1)=27^(x+1) for x. 9^(x1)=27^(x+1) => 9^x/ 9 = 27^x * 27 => 9^x / 27^x = 27*9 => 3^2x / 3^3x = 3^5 => 3^(2x  3x) = 3^5 As the base 3 is the same we...

Math
We have to solve for x given that : x^(4/3)1=15 x^(4/3)  1 = 15 => x^(4/3) = 15 + 1 => x^( 4/3) = 16 Take the log to base 4 of both the sides => log [ x^(4/3)] = log 4^2 => ...

Math
When you raise any square root to the 2nd power, you will get the values inside the square root by itself. For example: If 2= sqrtx , then, we know that x = 4. We can determine the value by...

Math
Using the law of exponentials yields: `a^6 = a^(2*3) => a^(2*3) = (a^2)^3` Reasoning by analogy yields: `b^(2*3) = (b^2)^3` Converting the difference of cubes into a product yields: `(a^2)^3 ...

Math
The algebraic form of an imaginary root is `z = x + i*y` , where x represents the real part and y represents the imaginary part. A polynomial can only have an even number of comeplex roots since if...

Math
Yest it is important. When we know that location of the angle, we can determine the values of the trigonometric identities. For example: We will compare the angle pi/3 into the first, second,...

Math
The value of cos 30 is usually known to all students. cos 30 = (sqrt 3)/2. Now we use the formula for cos 2x, which is: cos 2x = (cos x)^2  (sin x)^2 => cos 2x = 2 (cos x)^2  1 Now substitute...

Math
Let the distance where the defender and the prisoner met be D. Then the distance the defender ran = D Then, the distance the prisoner ran = D27 The number of steps that the defender ran to D is x...

Math
We have the vectors u = 3i  4j and v = 2i + j. Now the cosine of the angle between two vectors A = a1*i + b1*j and B = a2*i + b2*j is given by cos...

Math
You have to be a little bit more specific. If you are referring to the constraints of existence imposed by the denominators of the fraction, the value of the given expression has to be different...

Math
We need to find the antiderivative of f(x)=x*e^4x We solve this problem using Integration by parts: Int [f(x)g'(x) dx = f(x)g(x) Int [ f'(x)g(x) dx] Let f(x) = x and g'(x)= e^4x => g(x) =...

Math
We have a + b = 9 and a – b= 2. The easiest way to find a^4 + b^4 would be to first use the given values of a + b and a – b to determine a and b. a + b = 9 … (1) a – b = 2 … (2) (2) +...

Math
We have the function f(x) = cos 2x /(sin x)^2*(cos x)^2 cos 2x = (cos x)^2  ( sin x)^2 => cos 2x /(sin x)^2*(cos x)^2 => [(cos x)^2  ( sin x)^2] / (sin x)^2*(cos x)^2 => 1/ (sin x)^2 ...

Math
We have to find x given that 262x+1=52. 262x+1=52 cancel 26 from both the sides. => 2x + 1 = 2 Now x represents the absolute value of x and has the same value for x as well as x. =>...

Math
We have to solve : (1/5)[(1/5)(a 1/5)  (1/5)] = 2/5 and determine the value of a. (1/5)[(1/5)(a 1/5)  (1/5)] = 2/5 => [(1/5)(a 1/5)  (1/5)] = 2 open the brackets => (1/5)a  (1/5)(1/5)...

Math
We have to verify if cos^4x + 2sin^2x  sin^4x = 1. Now cos^4x + 2sin^2x  sin^4x => [(cos x)^2]^2 + 2sin^2x  sin^4x Now use the relation (cos x)^2 = 1  (sin x)^2 => [1  (sin x)^2]^2 +...

Math
We have the curve y=x^2  2x + ln e^2. Now we need to find the extreme point of the curve. For this, we first find the first derivative of the curve. y = x^2  2x + ln e^2 => y = x^2  2x + 2...

Math
We have to solve 13^(13x+1)=3^x for x. Looking at 13^(13x+1)=3^x, the bases are not the same, so we cannot equate the exponential. Therefore, we can only solve the problem using logarithms. Take...

Math
We have to prove that 1/cos^2x+tan^2y=1/cos^2y+tan^2x Now 1/( cos x)^2 + ( tan y )^2  1/ (cos y)^2  (tan x)^2 => [(sin y)^2/ (cos y)^2] + 1/ (cos x)^2  1/ (cos y)^2  [(sin x)^2/ (cos x)^2]...

Math
We'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)....

Math
We 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...

Math
To 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 ...

Math
Given 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...

Math
Given 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 ==>...

Math
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)...

Math
Given 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) =...

Math
Given 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...

Math
Given 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...

Math
a = ( 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 /...

Math
Let 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...

Math
Given 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) ==>...

Math
1/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...

Math
Given the point (2, k) and the line 3x+3y=4 Use the point (2, k) and let x=2 and y=k, and plug into the line 3x+3y=4. 3(2) + 3(k) = 4 Solve for k. 6 + 3k =4 3k=46 3k=2 k=2/3

Math
Let 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...

Math
The 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...

Math
Given 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 ...

Math
Let 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...

Math
I 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...

Math
Let 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...