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Solving Math Problem

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Solving math problem is a very helpful tool to get answers to calculus, algebra, trigonometry or any other topic you are having trouble with. Mathematics is the abstract study of topics. This page is about solving math problems and solutions which will let you know about this concept with clear understanding. This page gives a detailed solution of some of the topics in mathematics as many students find it very difficult to solve math problems. Students can learn to solve algebra problems, percentage problems, vectors etc.  

Solving Algebra Problems

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Algebra is a branch of mathematics which helps in finding the value of the unknown variables written in combination of terms.
It is the foundation stone of mathematics. Letters and symbols are known as variables.
Examples : x + 7 = 12, xyz + 37xy + z = 0, when an equation contains variables, you will often have to solve for one of those variables.

Example 1: Solve 7x + 120 = 0.
Solution:
Given equation is 7x + 120 = 0
Put all the variables aside and value on the other side.
7x = - 120
Divide both sides by 7, to find the value of 'x'.
$\frac{7x}{7}$ = − $\frac{120}{7}$
x = 17.14

Example 2 : Solve 5x$^{2}$ + 12x + 8 = 0
Solution: Compare given equation with quadratic general equation, we get
 a =  5,
b = 12 and
c = 8
Quadratic equation formula :
x = $\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
Substitute the values of a, b and c in the above formula
 x = $\frac{-12\pm\sqrt{12^{2}-4*5*8}}{2*5}$

 x = $\frac{-12\pm\sqrt{-16}}{10}$

There exist two values for x, let they be $x_{1}$ and x$_{2}$

x$_{1}$ = $\frac{-12+\sqrt{-16}}{10}$

= - $\frac{6}{5}$ + $\frac{2}{5}$ i

x$_{2}$ =  $\frac{-12-\sqrt{-16}}{10}$

= - $\frac{6}{5}$ - $\frac{2}{5}$ i

Therefore the values of x$_{1}$ and x$_{2}$ are  - $\frac{6}{5}$ + $\frac{2}{5}$ i and - $\frac{6}{5}$ - $\frac{2}{5}$ i.

Example 3:  By using completing square method solve 4x$^{2}$ + 3x + 7 = 0.
Solution:
 Through out the equation divide by 4 as the given equation is not in standard form.
x$^{2}$ + $\frac{3}{4}$x + $\frac{7}{4}$ = 0

To the right move all the constant terms
x$^{2}$ + $\frac{3}{4}$x = - $\frac{7}{4}$

coefficient of x term = $\frac{3}{4}$

Square half of the coefficient of x term, $\frac{9}{64}$

Add  $\frac{9}{64}$  to both sides

x$^{2}$ + $\frac{3}{4}$x + $\frac{9}{64}$ = - $\frac{7}{4}$ + $\frac{9}{64}$

x$^{2}$ + $\frac{3}{4}$x + $\frac{9}{64}$ = - $\frac{103}{64}$

On to the left we write the perfect square
(x + $\frac{3}{8}$)$^{2}$ = - $\frac{103}{64}$

On both sides take the square root
x + $\frac{3}{8}$  = $\pm\sqrt{-\frac{103}{64}}$

Solve for x
x = -$\frac{3}{8}$ $\pm\sqrt{-\frac{103}{64}}$

or x = - $\frac{3}{8}$ $\pm$ $\frac{1}{8}$ $\sqrt{103}$i

Solving Percentage

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Percentage is a number as a fraction of 100. In a mathematical language percent means out of 100. They are used in media to describe difference in interest rates, VAT, success rate, exam results etc.,
To change a fraction to a percentage, divide the numerator by the denominator and multiply by 100%.

Example 1: In a playground there are 2,000 people. Out of them 825 are players. Find the percentage of players in the class?

Solution:
Number of people in the playground = 2,000
Players in the playground = 825
To find percentage of players

= $\frac{825}{2000}$ * 100

= 41.25%

Example 2 : What is the percentage change in the price of a shoe if it had dropped down to $\$$110 from $\$$130?
Solution:
Percentage change formula is given as,
Percentage Change = $\frac{Old\ Value - New\ Value}{Old\ Value}$$\times$ 100
Percentage change in the price of a shoe
= $\frac{130 - 110}{130}$ * 100

= $\frac{20}{100}$ * 100
= 15.38 %

 Example 3: 47 is what percent of 90?
Solution: The given problem is solved by dividing $\frac{47}{90}$ and then the obtained result is multiplied by 100. The final solution should be followed with the "%" sign.

$\frac{47}{90}$ * 100

= 0.522 * 100
= 52.2
Therefore, it is 52.2%.

Solving Vectors

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A vector is a quantity having both magnitude and direction. Magnitude of a vector is known as its length. Examples of vectors are velocity, weight, force etc., If a is a vector, then its length is denoted by determinant of a or |a|.
Vectors are represented by thick letters or letter with bar. A vector can be represented using an ordered set of components in such a way that V = (V$_{1}$, V$_{2}$,......,V$_{n}$).

Given below are some of the example problems on vector:
Example 1: Find the length of the vector 9i + 2j + 4k.
Solution:
Let x = 9, y = 2, z = 4
Formula to find the length of the vector is $\sqrt{x^{2}+ y^{2}+ z^{2}}$Plug in the values in the above formula
$\sqrt{9^{2}+ 2^{2}+ 4^{2}}$
= 10.05

Example 2: Find the vector projection of $\vec{u}$ = 5$\vec{i}$ - 2$\vec{j}$ on the vector $\vec{v}$ = 5$\vec{i}$ + 8$\vec{j}$.
Solution:
$\vec{u}$ = 5$\vec{i}$ - 2$\vec{j}$ = 5 (1, 0) - 2 (0, 1)
= (5, 0) + (0, -2)  = (5, -2)

$\vec{v}$ = 5$\vec{i}$ + 8 $\vec{j}$ = 5 (1, 0) + 8 (0, 1)
= (5, 0) + (0, 8) = (5, 8)
 Vector projection formula
proj$_{v}$ u = $\frac{\vec{u}.\vec{v}}{\vec{|v|}^{2}}$$\vec{v}$
$\frac{7.5 - 2.8}{25+64}$ (5, 8)

= $\frac{35 - 16}{89}$ (5, 8)

= ($\frac{95}{89}$, $\frac{152}{89}$)

= (1.07, 1.71)