Showing posts with label math help. Show all posts
Showing posts with label math help. Show all posts

Thursday, August 20, 2009

Completing the Square Method

Sometimes the roots of a quadratic equation cannot be obtained by simple factorization. So, a more general method is used. This method, which is based on the fact that any quadratic equation may be written in the form of (x+p)2 = q, where p and q are real numbers, is known as completing the square method.

Question:-

Solve the equation by completing the square method.

2x2-7x+9=(x-3)(x+1)+3x

Answer:-

2x2-7x+9 = x2+x-3x-3+3x
-x2-x+3 -x2-x+3
---------------------------
x2-8x+12 = 0

now we solve this quadratic equation by completing the square method

x2-8x=-12

x2-8x+(8/2)2=(8/2)2-12

x2-8x+16 = 16-12

(x-4)2 = 4

taking the square root on both sides, square root symbol looks like √ .

√(x-4)2= √4

x-4 = ±2

We also can use square root calculator to get these values.

x-4 = +2 or x-4 = -2

x=6 or x=2 is the answer

This equation have just one variable.similarly we can also work on linear equations in two variables

Sunday, August 16, 2009

Problem on Finding Area of Rectangle

Topic : Area of rectangle
The area of a square is denoted by the formula Area = width * height
To calculate the area of a plane figure we use formulas such as area of triangle, square ,rectangles , or circles.

Here is a simple example for you
Question:
The cost of enclosing a rectangle garden with fence all round at rate 75 paisa / meter is Rs 300. If the length of the garden is 120 meters. Find the area of the garden in square meters.

Answer:









We know that,

Cost = Rate x Perimerter (as fencing is done at perimeter)
Let the width of garden be "w"
300 = 0.75 * 2(w + 120)
300 = 1.50 (w + 120)
divide both sides by 1.5
300/1.5 = 1.5/1.5(w+120)
200 = (w+120)
w+120 =200
subtract 120 from both sides
w = 80,

so width = 80 meters

Area of the feild = length *width
= 80 *120
= 9600 m2

Sunday, May 24, 2009

Question on Limits and Convergence of Number Sequence

In Number theory tutorialoffered by TutorVista will help you to solve problem related to number sequences, calculating limits and continuity of the sequence.

Topic : Limits and convergence of given number sequence.

Decrease in the value of the numbers in the sequence is said to be sequence convergence.

Question : Write the first five terms of the sequence, determine whether the sequence converges, and if so find its limit.

(1 - \frac{1}{2}), (\frac{1}{2} - \frac{1}{3}), (\frac{1}{3} - \frac{1}{4}), (\frac{1}{4} - \frac{1}{5}), . . . . . .



Solution :


General term will be given as ,
(\frac{1}{n}-\frac{1}{n+1})<br />\\ a_ =  \frac{1}{1} - \frac{1}{1+1} = 1 - \frac{1}{2} = \frac{2-1}{2} = \frac{1}{2} = 0.5 \\ a_2 =  \frac{1}{2} - \frac{1}{3} =  \frac{3-2}{2*3} = \frac{1}{6} = 0.167 \\a_3=  \frac{1}{3} - \frac{1}{4} =  \frac{4-3}{3*4} = \frac{1}{12} = 0.083 \\ a_4 =  \frac{1}{4} - \frac{1}{5} =  \frac{5-4}{5*4} = \frac{1}{20} = 0.05 \\ a_5 = \frac{1}{5} - \frac{1}{6} =  \frac{6-5}{6*5} = \frac{1}{30} = 0.03


















So 0.5, 0.167, 0.083, 0.05, 0.03 Thus we can observe that the values goes on decreasing.
Hence the number sequence converges.

Nows lets find it's limit by ratio test

\mathop{\lim}\limits_{n \to \infty}(\frac{a_{n+1}}{a_n}) \\ = \mathop{\lim}\limits_{n \to \infty}(\frac{\frac{1}{n+1} -\frac{1}{n+1+1}}{\frac{1}{n} -\frac{1}{n+1}}) \\ = \mathop{\lim}\limits_{n \to \infty}(\frac{\frac{n+2-(n+1)}{(n+1)(n+2)}}{\frac{n+1-n}{n(n+1)}}) \\ \mathop{\lim}\limits_{n \to \infty} (\frac{n+2-n-1}{(n+1)(n+2)} \div \frac{n+1-n}{n(n+1)}) \\ \mathop{\lim}\limits_{n \to \infty}(\frac{1}{(n+1)(n+2)}X\frac{n(n+1)}{1}) \\ \mathop{\lim}\limits_{n \to \infty}(\frac{n}{n+2}) \\ = \frac{\infty}{\infty+2} \\ = \frac{\infty}{\infty} \\ = 1<br />

















For more help contact math help or calculus help.