Quadratic Equations

A quadratic equation in the variable of x is an equation of the general form ax^2 + bx + c = 0 where a, b and c are coefficients, a ≠ 0. The letter "x" is the variable or unknown.

The constants a is called coefficient of x^2, b is called linear coefficient (coefficient of x) and c is called the constant term.

Similarly, 2x^2 – 3x + 1 = 0, 3x^2-5x -2 = 0 and x^2-x- 300 = 0 are also quadratic equations. In fact, any equation of the form p(x) = 0.

Where p(x) is a polynomial of degree 2, is the quadratic equation. This is the simplest way to the factorize would be to find the common factors.

Solving quadratic equation using quadratic formula:

X = [-b ± √(b^2 - 4ac) ]/ 2a

Solve the quadratic equation using factoring method:

Example problem 1:

Solve: x^2-10x + 9=0

Solution:

Factoring by splitting the middle term method,

Here a = coefficient of x^2 = 1

b = coefficient of x = -10

c = constant term = 9

We find a × c = 1× 9 = 9 = -1*-9, (-1) + (-9) = -10 = b.

x^2 - 10x + 9 = 0

x^2 + (- 1 - 9)x + 9 = 0

x^2 – 1 x – 9 x + 9 = 0

x (x - 1) – 9 (x - 1) = 0

(x – 1) (x – 9) = 0

x = 1, 9

So, the answer is x = 1, 9.

Between, if you have problem on these topics Applications of Quadratic Equations, please browse expert math related websites for more help on online math equation solver.

Solve the quadratic equation using quadratic formula:

Example problem 2:

Solve: x^2-4x + 3=0

Solution:

Coefficients are: a = 1, b = -4, c = 3

Quadratic Formula: x = [-b ± √ (b^2 - 4ac)] / 2a

Put in a, b and c: x = [-(-4) ± √ [(-4)2-4×1× (3)] / (2×1)

Solve:  x = [4 ± √ (16-12)]/2

x = [4 ± √ (4)]/2

x = (4 ± 2)/2

x= (4+2)/2                                       x = (4-2)/2

x= (6)/2                                           x= (2)/2

x=3                                                  x=1

So, the answer is x=3, 1.

Practice problems on quadratic equations:

1)      Solve the quadratic equation using quadratic formula: x^2 +10x + 9=0 (Answer: x = -1, -9).

2)      Solve the quadratic equation using quadratic formula: x^2+ 5x + 4=0 (Answer: x = -1, -4).

how to measure diameter of a circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane which is equidistant from a given point called the center. The common distance of the points of a circle from its center is called its radius. Circles are simple closed curves which divide the plane into two regions, an interior and an exterior. The way of measuring the diameter of the circle is nothing but finding the diameter of the circle.



(Source: Wikipedia)

I like to share this Diameter of a Circle Formula with you all through my article.

(i) How to measure diameter of a circle - from circumference of the circle

We know that the circumference of the circle, c = `pi` d

Where, d is the diameter of the circle.

Therefore, d = `c / pi`

(ii) How to measure diameter of a circle - from diameter of the circle:

Area of the circle, a = `pi /4 d^2`

Diameter of the circle, d = 2  `sqrt(a / pi)`

Example problems for “How to measure diameter of a circle”:

1. Find the diameter of the circle of circumference 50 cm

Solution:

Let c = 50cm

To find: Diameter of circle, d

Formula:

d = `c /pi`

Substitution of values:

d = `50 / pi`

= 15.92 cm

Between, if you have problem on these topics Area of a Circle Formula, please browse expert math related websites for more help solving math word problems.

2. Find the diameter of the circle of area 50 cm

Solution:

Let a = 50cm

To find: Diameter of circle, d

Formula:

d = 2  `sqrt(a / pi)`

Substitution of values:

d = 2  `sqrt(50 / pi)`

= 2 `sqrt(15.92)`

= 7.98 cm

3.Find the diameter of the circle of area 25 cm

Solution:

Let a = 25cm

To find: Diameter of circle, d

Formula:

d = 2  `sqrt(a / pi)`

Substitution of values:

d = 2  `sqrt(25 / pi)`

= 2 `sqrt(7.96)`

= 5.64 cm

Practice problems for “How to measure diameter of a circle”

1. Find the diameter of the circle of circumference 7cm
2. Find the diameter of the circle of circumference 22 cm

Solution for practice problems for “How to measure diameter of a circle”:

1. 22 cm
2. 14 cm

Solve for Real Numbers

Let us study about solve for real numbers. In mathematics real numbers express themselves by containing both rational and irrational numbers in it.

The numbers which can express themselves in ratio as b/c where ‘c’ is not equal to ‘0’ are termed as rational numbers.

The numbers which cannot express themselves in the form of ratio are termed as irrational numbers. Some of the examples are discussed in detail as below.


Solve for real numbers:

Example 1:

John has Rs. 6000 in his account. Then he gave 1/3 of the amount to his friend. Solve this to find how much amount John has in his account currently?

Solution:

Given:

John has Rs. 6000

He gave 1/3 of the amount to his friend

Then the amount given to John friend is calculated as follows:

= Rs. 6000 x 1/3.

= Rs. 2000 x 1

= Rs. 2000 (the amount he gave to his friend)

Therefore to get the remaining amount subtract the amount gave to his friend from the original amount as follows:

= Rs. 6000 – Rs. 2000

= Rs. 4000 (remaining amount that the John have currently and it is also real numbers)


Example 2:

Rama reached 2/5 km in 2 hours and Seema reached 1/3 km in 2 hours. Solve this to find the difference in the distance among them?

Solution:

Given:

Rama reached 2/5 km in 2 hours

Seema reached 1/3 km in 2 hours

Then the differences in the distance among the two are calculated as follows:

= 2/5 – 1/3 (subtract the values)

= (2/5) (3/3) – (1/3) (5/5) (multiply and divide one of them with ‘3’ and the other with ‘5’ in order to get common divisor for both the ratio’s)

= 6/15 – 5/15

= (6-5)/15 (take the number ‘15’ as the common divisor from both the ratio’s)

= 1/15 (it is also real numbers)

Between, if you have problem on these topics The Real Number System, please browse expert math related websites for more help on solve my math problem for me.

Exercises:

1. Sudha is earning Rs. 81 per hour by preparing work lists. If she prepares for 2/5 hours then how much she will earn? (Answer: 30)

2. Rahul is having 1/3 meters of cloth, Srimathi is having 3/2 meters of cloth and Mano is having 2/3 meters of cloth. Find the total meters of cloth they have with them? (Answer: 5/2 meters)

Write Fraction as a Decimal

A division of a whole called fraction. Fractions can be creating of minimum two numbers. The numerator specified as top number. The denominator specified as bottom number.

Understanding What is a Fraction? is always challenging for me but thanks to all math help websites to help me out.

In mathematics, ten as it base called as decimal number system. The numbers using the base-10 numeral system specified as decimal notation. A dot with a decimal number, like to present in 0.287. Here we are going to see about write fraction as a decimal.

Examples problem for write fraction as a decimal:

1. Write fraction as a decimal, given fractions are i) `2/10` and ii) `9/10` .

Solution:

Given fractions, i) `2/10`

`= 2/10`

The 2 divided by 10, we get the decimal value is,

`= 0.2`

The fraction, `2/10` gives the decimal is 0.2 (two tenths).

Given fractions, ii) `9/10`

`= 9/10`

The 9 divided by 10, we get the decimal value is,

`= 0.9`

The fraction, `9/10` gives the decimal is 0.9 (nine tenths).

2. Write fraction as a decimals correct to two decimal places, given fractions are i) 75/20 and ii) 52/36

Solution:

Given fractions, i) `75/20`

`= 75/20`

The 75 divided by 20, we get the correct to two decimal places is,

`= 3.75`

The fraction, `75/20` gives the decimal is 3.75 (three and sevety-five hundredths).

Given fractions, ii) `52/36`

`= 52/36`

The 52 divided by 36, we get the correct to two decimal places is,

`= 1.44`

The fraction, `52/36` gives the decimal is 1.44 (one and forty-four hundredths).

3. Write fraction as a decimals, correct to three decimal places, given fractions are i) `88/48` and ii) `27/8`

Solution:

Given fractions, i) `88/48`

`= 88/48`

The 88 divided by 48, we get the correct to three decimal places is,

`= 1.833`

The fraction, `88/48` gives the decimal is 1.833 (one and eight hundred thirty-three thousandths).

Given fractions, ii) `27/8`

`= 27/8`

The 27 divided by 8, we get the correct to three decimal places is,

`= 3.375`

The fraction, `27/8` gives the decimal is 3.375 (three and three hundred seventy-five thousandths).

I am planning to write more post on Convert a Fraction to a Decimal and how to simplify large fractions. Keep checking my blog.

Practice problems for write fraction as a decimal:

1. Write fraction as a decimal, given fractions are i) `7/10` and ii) `4/20` .

Answer is,

i) 0.7,

ii) 0.4

2. Write fraction as a decimal correct to two decimal places, given fractions are i) `19/3` and ii) `38/12`

Answer is,

i) 6.33,

ii) 3.16

Finding Lowest Common Multiple

 In arithmetic and number theory, the lowest common multiple or (LCM) least common multiple or smallest common multiple of two rational numbers a and b is the smallest positive rational number that is an integer multiple of both a and b. (The definition may be extended to any two real numbers whose ratio is a rational number.) Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then LCM(a, b) is defined to be zero.- Source wikipedia

Understanding LCM Finder is always challenging for me but thanks to all math help websites to help me out.

Examples for finding lowest common multiple:

Example 1:

Find lowest common multiple for the number 24 and 34.
On finding the LCM we have to find the prime factors for the given number.
24      : 2 2 2 3
34      : 2          17
---------------------------
LCM   : 2 2 2 3 17
Lowest common multiple for the numbers 24 and 34 is 2 * 2 * 2 * 3 * 17 = 408
Example 2:
Find lowest common multiple for the number 44 and 54.
On finding the LCM we have to find the prime factors for the given number.
44      : 2 2 11
54      : 2         3 3 3
----------------------------
LCM   : 2 2 11  3 3 3
Lowest common multiple for the numbers 44 and 54 is 2 * 2 * 11 * 3 * 3 * 3 = 1188
Example 3:
Find lowest common multiple for the number 64 and 84.
On finding the LCM we have to find the prime factors for the given number.
64      : 2 2 2 2 2 2
84      : 2 2            3 7
---------------------------------
LCM  :  2 2 2 2 2 2 3 7
Lowest common multiple for the numbers 64 and 84 is 2 * 2 * 2 * 2 * 2 * 2 * 3 * 7= 1344

Between, if you have problem on these topics How to Find LCM, please browse expert math related websites for more help on neet exam 2013 syllabus.

Practice problems for finding lowest common multiple:

Problem 1:
Find lowest common multiple for the number 242 and 342.
Lowest common multiple for the numbers 242  and 342 is 41382
Problem 2:
Find lowest common multiple for the number 224 and 324.
Lowest common multiple for the numbers 224 and 324is 18144
Problem 3:
Find lowest common multiple for the number 244 and 344.
Lowest common multiple for the numbers 244 and 344 is 20984