Given:

a + 2b = 10 and 2ab = 9

Formula used:

(a + b)² = a² + b² + 2ab

(a - b)² = a² +b² – 2ab

Calculation:

On squaring both sides of given equation we get,

(a + 2b)² = (10)²

⇒ (a² + 4b² +4ab) = 100

⇒ a² + 4b² + 18 = 100  

⇒ a² + 4b² = 82

Now,

(a – 2b)^{2 =} a² +4b² -4ab

Substituting values of a² + 4b² and 4ab from above

We get,

(a – 2b)² = 82 – 18

⇒ (a – 2b)^{2 =} 64

⇒ (a – 2b) = ±8

⇒ |a – 2b| = 8

∴ The value of |a – 2b| is 8

Calculation:

3x(x – 1) – 7x2 

⇒ 3x × x – 3x × 1 – 7x² 

⇒ 3x² – 3x – 7x² 

∴ –4x² – 3x

Given:

8a3 + b3 = 16 and 2a+ b = 4

Formula:

a³ + b³ = (a + b)(a² - ab + b²)

(a + b)² = (a² + 2ab + b²)

Calculation:

⇒ (8a³ + b³) = (2a + b)(4a² - 2ab + b²)

⇒ 16 = 4 × (4a2 - 2ab + b2)

⇒ 4a2 - 2ab + b2 = 4

⇒ (2a + b)² = 4a² + 4ab + b²

⇒ 16 = 4a2 + 4ab + b2

Solving,

⇒ 6ab = 12

⇒ ab = 2

Then,

⇒ 4a² + b² = 8

Then,

⇒ ? = 16a4 + b4

⇒ ? = (4a² + b²)² - 8a²b²

⇒ ? = 8² - 8 × (2)²

⇒ ? = 32

∴ 16a4 + b4 = 32

Given :

 27(x + y)3 - 8(x - y)3 = (x + 5y)(Ax2 + By2 + Cxy)

Formula used :

P³ - q³ = (p - q) (p² + q² + pq)

Calculations :

27(x + y)³ - 8(x - y)³ = [3(x + y)]³ - [2(x - y)]³

we use given formula above

⇒ [3(x + y) - 2(x - y)]  [(3x + 3y)² + (2x - 2y)² + 3(x + y) × 2(x - y)]

⇒ (x + 5y) (19x² + 7y² + 10xy)

Now compare (x + 5y) (19x2 + 7y2 + 10xy) with (x + 5y)(Ax2 + By2 + Cxy)

We will get A = 19, B = 7 and C = 10 

So, 

A + B - C = 19 + 7 - 10 

⇒ 16 

∴ The value of A + B - C is 16

Given: 

x = √6 + √5

Formula used: 

(a - b)² = a² + b² - 2ab

Calculation:

x = √6 + √5

⇒ 1/x = √6 - √5 

⇒ x - 1/x = 2√5

⇒ (x - 1/x)² = (2√5)²

⇒ x² + 1/x² - 2 = 20

∴ The value of x2 + 1/x2 - 2 is 20.

Introduction to Algebra

Algebra is derived from the Arabic word “al-jabr” which means ‘collection of broken part’. It is regarded is block point for basics of mathematics. To make mathematics easy in dealing with calculations algebra is introduced. It acts as a tying rope between different section of mathematics.

One of the notable thing about algebra is that we read only elementary algebra which is basics of it. We read it in mathematics, science upto engineering study. For simple calculations, elementry is enough to deal with.

Professional’s (algebraist) read ‘Abstract algebra’ also known as ‘Modern algebra’ which is used to study Advanced mathematics at research level.

In real life situations it plays an important role:

Q.When we go to a shop and bought a toy worth ₹16 and gave ₹20 to shopkeeper then what amount he has to return or I have to pay?

Let’s come to the situation numerically,

$$\fbox{ }+{16}={20}$$

Here is a single variable so using the box is easier but we can't use the box everywhere as there are different variable.

So we started using ‘x’ in place of box. You can use any alphabet in place of it. Such as, if there is more than one variable we use x, y, or any alphabet you want.

Now,

x + 16 = 20

We can easily answer this as he will return ₹ 4 but we will go step by step to understand it.

Algebra rule means as a puzzle to enjoy. So you should also enjoy solving it.

Equality sign has a great significance as it is the one which tells you what amount you need to attach or detach from left-hand side and put it on right-hand side so there will be no effect on equality.

We have to find only ‘x’

16 is extra in left-hand side so we should subtract it also from right-hand side to maintain equality.

x + 16 – (16) = 20- (16)

x = 4

Thus, the shopkeeper will return ₹ 4 to us.

See this quadratic equation

ax² + bx + c = 0

as it contains many terms a, b, c as constant.

a is coefficient of x².

b is coefficient of x.

c is constant term.

Most important thing is ‘x’ which is variable and we have to find solution.

To find its solution there is a simple formula

$$X=\frac{-b±\sqrt{b^{2}-4ac}}{2a}$$

Will provide a maximum of two solutions.

Given

2a + 3a = 7a - 4a + 12

Concept

Separate the variable and non variable terms first and then solve the equation.

Calculation

2a + 3a = 7a - 4a + 12

5a = 3a + 12

5a - 3a = 12

2a = 12

a = 6

Given:

Sum of two numbers is 150

Let first number be 'a' and second number be 'b'

Calculation:

According to question, 

⇒ (a + b) = 150       (1)

Now,

One-fourth of a is 5 more than one-sixth of b 

⇒ {(a/4) - (b/6)} = 5

⇒ 3a - 2b = 60     (2)

Eq. (1) × 2 and eq. (2) × 1 multiply by and than add

⇒ {(2a + 2b = 300) + (3a - 2b = 60)}

⇒ 5a = 360

⇒ a = 72

∴ Smaller number is 72 

Concept used:
a³ + b³ = (a + b) × (a² – ab + b²)

Calculation:
\(\frac{17}{15} × \frac{17}{15} × \frac{17}{15}\space +\space \frac{15}{17} × \frac{15}{17} × \frac{15}{17} \over \frac{17}{15} × \frac{17}{15}\space -\space 1 \space+\space \frac{15}{17} × \frac{15}{17} \)
Let 17/15 and 15/17 be a and b respectively.
⇒ a × b = 17/15 × 15/17 = 1
⇒ (a × a × a + b × b × b)/(a × a – 1 + b × b)
⇒ (a³ + b³)/(a² – ab + b²)
⇒ [(a + b) × (a2 – ab + b2)]/(a2 – ab + b2)
⇒ a + b
Putting the value of a and b,
⇒ 17/15 + 15/17
⇒ (17 × 17 + 15 × 15)/(15 × 17)
⇒ (289 + 225)/255
⇒ 514/255
∴ The correct answer is 514/255.

An expression with one or more terms is known as a polynomial in general. For example: 5x, 2a + 3b, m + n – 3. While an expression that contains only one term is called a monomial. For example: 7xy, 2x, -4n, -8, 2 3ab 

• Division of polynomial by monomial: Consider a polynomial (9x2 – 15xy), and monomial 3x. On dividing, we get ,\((9x2 – 15xy)\over3x\) = \(3x(3x\ -\ 5y)\over3x\) = 3x - 5y
• Division of polynomial by polynomial: Consider two polynomial 11x + 15x2 - 12, and 5x - 3. If we were to divide them, we will write them as \(15x^2+11x - 12\over 5x - 3\)