Algebraic Formulas

Algebraic Formulas
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Algebra Formulas form the foundation of numerous topics of mathematics. Topics like equations, quadratic equations, polynomials, coordinate geometry, calculus, trigonometry, probability, extensively depend on algebra formulas for understanding and for solving complex problems. The algebra formulas are helpful to perform complex calculations in the least time and with fewer steps.

Based on the complexity of the math topics, the algebraic formulas have also been transformed. Topics like logarithms, indices, exponent, progressions, permutations, and combinations have their own set of algebraic formulas. Here, we shall look into the list of algebraic formulas used across the different math topics. 

Table of Contents

Algebraic Identities

An identity is an equation that is always true regardless of the values assigned to the variables. Algebraic Identity means that the left-hand side of the equation is identical to the right-hand side of the equation, and for all values of the variables. Algebraic identities find applications in solving the values of unknown variables. Here are some most commonly used algebraic identities:

Algebraic Identities

Let us look at the algebraic identity: (a + b)2 = a2 + 2ab + b2, and try to understand this identity in algebra and also in geometry. As a proof of this formula, let us try to multiply algebrically the expression and try to find the formula. (a + b)= (a + b) × (a + b) = a(a + b) + b(a + b) = a2 + ab + ab + b2. This expression can be geometrically understood as the area of the four sub figures of the below given square diagram. Further, we can consolidate the proof of the identity (a + b)2= a2 + 2ab + b2.

Geometric Representation of (a + b)2

Algebraic Formulas

An algebraic formula is an equation, a rule written using mathematical and algebraic symbols. It is an equation that involves algebraic expressions on both sides. The algebraic formula is a short quick formula to solve complex algebraic calculations. These algebraic formulas can be derived for each maths topic, having unknown variable x, and some of the common algebraic formulas can be applied in each of the maths topics.

Example:

(a+b)2 =a2 +2ab + b2 is an algebraic formula and here,

Algebra Formulas for Class 8 

The algebraic formulas for three variables a, b, and c and for a maximum degree of 3 can be easily derived by multiplying the expression by itself, based on the exponent value of the algebraic expression. The below formulas are for class 8. 

Algebraic Formulas for Class - 8
Some of the common laws of exponents with the same bases having different powers, and different bases having the same power, are useful to solve complex exponential terms. The higher exponential values can be easily solved without any expansion of the exponential terms. These exponential laws are further useful to derive some of the logarithmic laws.

Laws of Exponents

Algebra Formulas for Class 9

Logarithms are useful for the computation of highly complex multiplication and division calculations. The normal exponential form of 25 = 32 can be transformed to a logarithmic form as Log 32 to the base of 2 = 5. Further, the multiplication and division between two mathematic expressions can be easily transformed into addition and subtraction, after converting them to logarithmic form. The below properties of logarithms formulas, which are applicable in logarithmic calculations.  

Properties of Logarithms

Algebra Formulas for Class 10

An important algebra formula introduced in class 10 is the “quadratic formula”. The general form of the quadratic equation is ax2 + bx + c = 0, and there are two methods of solving this quadratic equation. The first method is to solve the quadratic equation by the algebraic method, and the second method is to solve through the use of the quadratic formula. The below formula is helpful to quickly find the values of the variable x with the least number of steps.

Quadratic Formula

In the above expression, the value b2 - 4ac is called the determinant and is useful to find the nature of the roots of the given equation. Based on the value of the determinant, the three types of roots are given below.

  • If b2 - 4ac > 0, then the quadratic equation has two distinct real roots.
  • If b2 - 4ac = 0,  then the quadratic equation has two equal real roots.
  • If b2 - 4ac < 0,  then the quadratic equation has two imaginary roots.

Apart from this, we have a few other formulas related to progressions. Progressions include some of the basic sequences such as arithmetic sequence and geometric sequence. The arithmetic sequence is obtained by adding a constant value to the successive terms of the series. The terms of the arithmetic sequence is a, a + d, a + 2d, a + 3d, a + 4d, .... a + (n - 1)d.  The geometric sequence is obtained by multiplying a constant value to the successive terms of the series. The terms of the geometric sequence are a, ar, ar2, ar3, ar4, .....arn-1.  The below formulas are helpful to find the nth term and the sum of the terms of the arithmetic, and geometric sequence.

Formulas for Geometric and Arithmetic Series

Algebra Formulas for Class 11

The important topics of Class 11 which have extensive use of algebraic formulas are permutations and combinations. Permutations help in finding the different arrangement of r things from the n available things, and combinations help in finding the different groups of r things from the available n things. The following formulas help in finding the permutations and combination values. 

Permutation and Combination Formulas

Apart from the permutations and combinations, there is another important topic of “Binomial Theorem” as well which is used to evaluate the large exponents of algebraic expressions with two terms. Here the coefficients of the binomial terms are calculated from the formula of combinations. The below expression provides the complete formula for binomial expansion, and it can be termed as the algebraic expression of the binomial theorem.

Binomial Expansion Formula

Algebra Formulas for Class 12

The vector algebra formulas that are involved in class 12 are as follows. 

For any three vectors, \( \vec{a}, \vec{b}\) and \(\vec{c}\):

  • The magnitude of \(\vec{a} = x \hat{i}+y \hat{j}+z \hat{k} \) is, \( |\vec{a}| = \sqrt{x^2+y^2+z^2} \).
     
  • The unit vector along \(\vec{a}\) is \(\dfrac{\vec{a}}{|\vec{a}|}\).
     
  • The dot product is defined as: \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta\), where \(\theta\) is the angle between the vectors \(\vec{a}\) and \(\vec{b}\).
     
  • The cross product is defined as: \(\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta \,\,\hat{n}\), where \(\theta\) is the angle between the vectors \(\vec{a}\) and \(\vec{b}\).
     
  • The scalar triple product of three vectors is given by \( [\vec{a} \text{  } \vec{b} \text{  } \vec{c} ] = \vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \cdot \vec{c} \).

Algebraic Function

An algebraic function is of the form y=f(x).  Here, x is the input and y is the output of this function. Here, each input corresponds to exactly one output. But multiple inputs may correspond to a single output. For example: f(x)= x2 is an algebraic function. Here, when x=2, f(2)= 22 =4. Here, x=2 is the input, and f(2)=4 is the output of the function.

Algebraic Functions

Algebraic Fractions

We can perform numerous arithmetic operations such as addition, subtraction, multiplication, and the dividing of fractions in algebra just the same way we do with fractions involving numbers. Further, it only has the unknown variables and involved the same rules of working across fractions. The below four expressions are useful for working with algebraic fractions. Adding Fractions:  x/y + z/w = (x.w + y.z)/(y.w)  

  1. Subtracting Fractions: x/y - z/w = (x.w - y.z)/(y.w)

  2. Multiplying Fractions: x/y × z/w  = xz/yw

  3. Dividing Fractions: x/y ÷ z/w = x/y × w/z = xw/yz

Tips and Tricks

The following quick tips and tricks would be useful to easily understand the memorize algebraic identities.

  1. You can memorize the algebraic identities by understanding how they are derived. For example, (a+b)2=(a+b)(a+b)= a2+ab+ab+b= a2+2ab+b2.
  2. In the same way, you can try to derive the other algebraic identities as well.

Challenging Questions

Now having understood the concepts of algebraic expression, check out the below three questions to better practice the learned concept.
  1. Find the roots of the quadratic equation: x2+7x+12=0
  2. Simplify the expression: (x-9y3)/(x-7y8) so that the answer has no negative exponents.
  3. Expand the logarithm: log x2y3 z.

Algebraic Formulas and Related Topics

Given below is the list of topics that are closely connected to Algebraic Formulas. These topics will also give you a glimpse of how such concepts are covered in Cuemath. 


FAQs on Algebraic Formulas

What are the Formulas In Algebra?

Here are some most commonly used formulas of algebra:

  • a2 - b2 =(a-b)(a+b)
  • (a+b)2 =a2 + 2ab + b2
  • (a-b)2=a2 - 2ab+b2
  • (x+a)(x+b)=x2+x(a+b)+a b 
  • (a+b+c)2=a2+b2+c2+2ab+2bc+2ca
  • (a+b)3 =a3 +3a2b+3ab2 +b3
  • (a - b)3 =a3 - 3a2b+3ab2 - b3

Further, there are algebraic formulas for other topics of maths such as exponents, logarithms, permutations, sequences, and vector algebra.

What Are the Basic Math Formulas?

Here are some basic math formulas: 

  • a2 - b2 =(a-b)(a+b)
  • (a+b)2 =a2 + 2ab + b2
  • (a-b)2 = a2 - 2ab + b2 
  • (x+a)(x+b)=x2 + x(a+b) + ab
  • (a+b+c)= a2 + b2 + c2 + 2ab + 2bc + 2ca
  • (a+b)3 =a3 +3a2b + 3ab+ b3
  • (a-b)3 =a3 - 3a2b + 3ab2- b3

How do I Learn Algebraic Formulas?

Algebraic formulas can be easily memorized by visualizing the formulas as squares or rectangles. Further, the understanding of the factorized forms of the formulas helps to easily learn and remember the algebraic formulas.

What Is The Formula For a2- b2?

The formula for a2- b2 is (a+b)(a-b)= a2- b2. It is called the difference of squares formula.

How to Solve Algebraic Formulas?

The solving of algebraic formulas is to aim at equalizing the left-hand side of the expression with the right-hand side of the expression. Further, the terms can be transferred from the left to the right side of the expression, based on the laws of algebra.

What is the Basis Of Algebraic Formulas?

The basis of algebraic formulas is that the resultant numeric value of the expressions on either side of the equals to sign is equal. Further, algebraically the terms are modified on either side to match up with the algebraic formulas.

What are Algebraic Expressions?

For each of the algebraic formulas, the equations with variables, powers, and arithmetic operations, and on either side of the equals to sign are called algebraic expressions. In the algebraic formula (a+b)(a-b)= a2- b2, the terms on either side of the equals too sign is called an algebraic expression.

What are the Algebraic Formula for Triangular Numbers? 

The algebraic formula for triangular numbers is H2 = B2 + A2  and it helps to relate the length of the sides of the triangle. It is applicable for a right triangle and has been derived from the Pythagaros theorem. The alphabets H represents the hypotenuse, B represents the base of the right triangle, and A represents the altitude of the triangle. Applying this same formula an example of triangular numbers is (6, 8, 10).

Algebraic Formulas Solved Examples

Here are a few problems on algebra along with their solutions for you.

Example 1: Using algebraic identities, find (2x-3y)2.

Solution:

Here, we use the identity (a-b)2 = a2 - 2ab + b2 to expand this. Here, a= 2x and b=3y. Then we get: (2x-3y)2 = (2x)2 -2(2x)(3y)+(3y)= 4x2 -12xy + 9y2. Therefore, (2x - 3y)2 = 4x2 -12xy + 9y2.

Example 2: Using identities, evaluate 297 × 303. 

Solution:

The above product can be written as (300-3) × (300+3). We will find this product using the formula: (a-b)(a+b)=a2- b2 Here a=300 and b=3. Then we get: (300-3) × (300+3) =3002 - 32 = 90000-9 = 89991. Therefore, 297 × 303 = 89991.

Example 3: Find the roots of the quadratic equation x2+5x+6=0 using the quadratic formula.

Solution:

The given equation is x2+5x+6=0. Comparing this with ax2+bx+c=0, we get: a=1;  b=5;  c=6. Substituting these values in the quadratic formula: 

\[ \begin{align}
x&=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\\[0.2cm]
x&=\frac{-5 \pm \sqrt{5^2-4 \cdot 1 \cdot 6}}{2 \cdot 1}\\[0.2cm]
x&=\frac{-5 \pm \sqrt{1}}{2 \cdot 1}\\[0.2cm]
x&=\frac{-5 \pm 1}{2}\\[0.2cm]
x &= \frac{-5+1}{2}, \,\,\, x=\frac{-5-1}{2}\\[0.2cm]
x &= -2; \,\,\, x=-3
\end{align} \]

Therefore x = -2, and -3.

Practice Questions on Algebraic Formulas

Here are a few problems related to algebraic formulas. Select/Type your answer and click the "Check Answer" button to see the result. 

 
 
 
 
 

 

 

 

 

  
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