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May 3, 202139 Insightful Publications

**Algebra Formulas:** Algebra is a field of mathematics that aids in solving mathematical equations and calculating unknown quantities such as variable values, constants, and percentages. Algebra is used when both fixed and dynamic components are present at the same time to determine a certain situation. In an algebra formula chart, Elementary Algebra, Advanced Algebra, Abstract Algebra, Linear Algebra, and Commutative Algebra are the different branches of algebra.

In this article, we will talk about all the algebra theorems like binomial theorem formula, the algebra formula chart etc. Algebra Formulas are necessary for students to excel in academics, and they must know all the algebra formulas. It is a crucial topic for students who want to score 90+ in their higher grades. With the help of Algebra, students learn how to find the unknown value in an equation. continue reading this article and learn all algebra formulas easily.

**Algebra** is generalised arithmetic in which we represent the numbers by letters, known as literals numbers or simply literals.

These letters do not have any fixed values and are called variables. In our real-life scenario, we see some values that keep on varying. However, there is a constant need to represent these fluctuating values.

Here in algebra, these values are often depicted with letters such as \( a,b,c,x,y,z,p,\) or \(q,\) and these letters are called variables. Now, what is the equation? These letters or values are influenced through various arithmetic operations of addition, subtraction, multiplication, and division to find the values. Using these arithmetic operations and the letters, we can form algebraic expressions or equations.

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**Definition:** The combination of the constants and the variables connected by some or all of the four fundamental operations addition \((+)\) subtraction \((-)\) multiplication \(\left( \times \right)\) and division \(\left( \div \right)\) is known as an algebraic expression.

**Examples: **\(4x + 5,10y – 5\) are examples of algebraic expressions.

**Definition: **The algebraic identities are the algebraic equation, which is valid for all the variables’ values. Algebraic equations are math expressions that include numbers, variables (unknown values) and mathematical operations (addition, subtraction, multiplication and division). For example, the binomial theorem formula.

Algebraic identities are used in different branches of maths, like algebra, geometry, trigonometry etc. These are mainly used to find the factors of the polynomials.

*Other important Maths formulas:*

If an equation is valid for all values of the variables in it, it is called an identity. The algebraic identities are the equation in which the value of the left-hand side of an equation identically equals the value of the right-hand side of the equation.

Example: Consider the linear equation \(ax + b = 0\)

Here, the left-hand side and right-hand sides of the above equations are the same when \(-\frac{b}{a}\)

The algebraic formulas for the three variables \(a,b\) and \(c\) and a maximum degree of \(3\) can be quickly taken by multiplying the expression by itself, based on the exponent value of the algebraic expression. You can see the basic algebraic formulas given below:

1. \({(a + b)^2} = {a^2} + 2ab + {b^2}\)

2. \({(a – b)^2} = {a^2} – 2ab + {b^2}\)

3. \((a + b)(a – b) = {a^2} – {b^2}\)

4. \({(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}\)

5. \({(a – b)^3} = {a^3} – 3{a^2}b + 3a{b^2} – {b^3}\)

6. \({a^3} + {b^3} = (a + b)\left( {{a^2} + ab + {b^2}} \right)\)

7. \({a^3} – {b^3} = (a – b)\left( {{a^2} + ab + {b^2}} \right)\)

8. \({(a + b + c)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca\)

A few of the ordinary laws of exponents with the similar base having different powers, and various bases having the same power, are utilised to solve the complex exponential terms. Students can compute the higher exponential values without the expansion of the exponential terms easily. These exponential laws are further helpful to derive some of the logarithmic laws.

(i) \({a^m} \cdot {a^n} = {a^{m + n}}\)

(ii) \(\frac{{{a^m}}}{{{a^n}}} = {a^{m – n}}\)

(iii) \({\left( {{a^m}} \right)^n} = an\)

(iv) \({(ab)^m} = {a^m} \cdot {b^m}\)

(v) \({a^0} = 1\)

(vi) \({a^{ – m}} = \frac{1}{{{a^m}}}\)

We can simplify the complex multiplication and division calculations by using Logarithms. The normal, exponential form of \({2^5} = 32\) can be transformed to a logarithmic form as \(32\) to the base of \(2\,=5\).

Students can easily tranform the complex multiplication and divison between two mathematic expressions into addition and subtraction with Logarithm. Below we have provided the properties of Logarithm formulas that students can use for logarithm calculations:

1. \({x^m} = a \Rightarrow {\log _x}a = m\)

2. \({\log _a}1 = 0\)

3. \({\log _a}a = 1\)

4. \({\log _a}(xy) = {\log _a}x + {\log _a}y\)

5. \({\log _a}\frac{x}{y} = {\log _a}x – {\log _a}y\)

6. \({\log _a}\left( {{x^m}} \right) = m{\log _a}x\)

7. \({\log _a}x = \frac{{{{\log }_c}x}}{{{{\log }_c}a}}\)

8. \({a^{\log ax}} = x\)

These formulas are essential to solving algebraic equations’ questions and play an essential role in solving questions of coordinate geometry, trigonometric functions, areas and perimeter, etc. The formulas are as given below:

1. \({a^2} – {b^2} = (a – b)(a + b)\)

2. \({(a + b)^2} = {a^2} + 2ab + {b^2}\)

3. \({a^2} + {b^2} = {(a – b)^2} + 2ab\)

4. \({(a – b)^2} = {a^2} – 2ab + {b^2}\)

5. \({(a + b + c)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2ac + 2bc\)

6. \({(a + b + c)^3} = {a^3} + {b^3} + {c^3} + 3(a + b)(b + c)(c + a)\)

7. \({a^3} + {b^3} + {c^3} – 3abc = (a + b + c)\left( {{a^2} + {b^2} + {c^2} – ab – ac – bc} \right)\)

8. \({(a – b – c)^2} = {a^2} + {b^2} + {c^2} – 2ab – 2ac + 2bc\)

9. \({(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3};{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\)

10. If \(a + b + c = 0\) then \({a^3} + {b^3} + {c^3} = 3abc\)

11. \({x^2} + x(a + b) + ab = (x + a)(x + b)\)

12. \(ab(a + b) + bc(c + b) + ca(c + a) = (a + b)(b + c)(c + a)\)

13. \({a^2}(b + c) + {b^2}(c + a) + {c^2}(a + b) + 3abc = (a + b + c)(ab + bc + ca)\)

14. \({a^2}(b – c) + {b^2}(c – a) + {c^2}(a – b) = (a – b)(b – c)(c – a)\)

15. If \(n\) is a natural number \({a^n} – {b^n} = (a – b)\left( {{a^{n – 1}} + {a^{n – 2}}b + \cdots + {b^{n – 2}}a + {b^{n – 1}}} \right)\)

16. If \(n\) is even \((n = 2k),{a^n} + {b^n} = (a + b)\left( {{a^{n – 1}} – {a^{n – 2}}b + \cdots + {b^{n – 2}}a – {b^{n – 1}}} \right)\)

17. If \(n\) is odd \((n = 2k + 1),{a^n} + {b^n} = (a + b)\left( {{a^{n – 1}} – {a^{n – 2}}b + \ldots – {b^{n – 2}}a + {b^{n – 1}}} \right)\)

18. \({(a + b + c + ..)^2} = {a^2} + {b^2} + {c^2} + \cdots + 2(ab + ac + bc + \cdots \)

(i) \(\left( {{a^m}} \right)\left( {{a^n}} \right) = {a^{m + n}}\)

(ii) \({(ab)^m} = {a^m}{b^m}\)

(iii) \({\left( {{a^m}} \right)^n} = {a^{mn}}\)

(iv) If \(a + b + c = abc\)

Besides addition, subtraction, and multiplication, you can see some other formulas of vectors in algebra. They are given below:

Let \(P(x,y,z)\) be the point. The vector position of \(P\) is \(\overrightarrow {OP} = \vec r = x\hat \imath + y\hat \jmath + z\hat k\) and the magnitude of the vector is shown by \(|\overrightarrow {OP\mid } = |\overrightarrow {r\mid } = \sqrt {{x^2} + {y^2} + {z^2}} \)

In case of any vector, \(r\) is the magnitude, \((l,m,n)\) are the directions confines and \((a,b,c)\) are the directions ratios, then you see:

\(l = \frac{a}{r},m = \frac{b}{r},n = \frac{e}{r}\)

Now, let \(|\vec a|\) and \(\mid \overrightarrow {b\mid } \) respectively in the ratio \(m:n\) internally is shown by \(\frac{{n\vec a + m\vec b}}{{m + n}}\)

Similarly, in the external division, the formula will be \(\frac{{m\vec b – n\vec a}}{{m – n}}\)

Two vectors of cross-product in matric representation is shown by:

\(\vec a \times \vec b = \left| {\begin{array}{*{20}{c}}{\hat \imath }&{\hat \jmath }&{\hat k}\\{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\end{array}} \right|\)

The types of all algebra formulas:

The quadratic equation can be written in the form of \(a{x^2} + bx + c = 0\), and there are a couple of methods to solve the quadratic equation. The algebraic method is the first method, and the quadratic formula is the second method used to solve quadratic equations. The below-given formula is useful to quickly find the values of the variable \(x\) with the least number of steps.

Quadratic formula: For \(a{x^2} + bx + c = 0\)

\(x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}\)

In the above expression, the value \({b^2} – 4ac\) is known as the determinant and is helpful to find the nature of the roots of the given equation. By looking at the value of the determinant, the three types of roots are given below:

– If \({b^2} – 4ac > 0\) then the quadratic equation will have two distinct real roots.

– If \({b^2} – 4ac = 0\) then the quadratic equation will have two equal real roots.

– If \({b^2} – 4ac < 0\) then the quadratic equation will have two imaginary roots.

The arithmetic sequence is acquired by adding a constant value and the successive terms of the series. The terms of a arithmetic sequence are given by \(a,a + d,a + 2d,a + 3d,a + 4d, \ldots ,a + (n – 1)d.\)

The geometric sequence is acquired by multiplying the constant value and successive terms of series. The terms of the geometric sequence are given by \(a,ar,a{r^2},a{r^3},a{r^4}, \ldots \ldots a{r^{n – 1}}\)

The below-given formulas are useful to identify the \({{\rm{n}}^{{\rm{th}}}}\) term and the sum of the terms of the arithmetic and geometric sequence.

1. \({{\rm{n}}^{{\rm{th}}}}\) term of an arithmetic sequence; \({a_n} = a + (n – 1)d\)

2. Sum of \(n\) terms on of an arithmetic sequence; \({S_n} = \frac{n}{2}(2a + (n – 1)d)\)

3. \({{\rm{n}}^{{\rm{th}}}}\) term of a geometric sequence \({a_n} = a \cdot {r^{n – 1}}\)

4. Sum of \(n\) terms of a geometric sequence; \({S_n} = \frac{{a\left( {1 – {r^n}} \right)}}{{1 – r}},r \ne 1\)

5. Sum of the infinite terms of a geometric sequence \(S = \frac{a}{{1 – r}}\)

Permutations help in finding the various arrangement of \(r\) things from the \(n\) available things, and the combinations help in finding the different groups of \(r\) things from the available \(n\) things.

The below-given formulas help us in finding the permutations and the combined values.

1. \(n! = n \times (n – 1) \times \ldots .3 \times 2 \times 1\)

2. \(nCr = \frac{{n!}}{{(n – r)!r!}}\)

3. \(nPr = \frac{{n!}}{{(n – r)!}}\)

\((x+y)^n=C_0x^ny^0+C_1x^{n-1}y^1+C_2x^{n-2}y^2+…+C_{n-1}xy^{n-1}+C_nx^0y^n\)

The general linear equation is represented as

\({a_1}{x_1} + {a_2}{x_2} \ldots \ldots \ldots + {a_n}{x_n} = b\)

Here,

(i) (\(a\) is representing the coefficients

(ii) \(x\) is representing the unknowns

(iii) \(b\) is representing the constant

There is the system of linear algebraic equations which is a set of equations. The system of equations can be calculated by using the matrices.

The linear function is shown below:

\(\left( {{x_1}, \ldots \ldots ..{x_n}} \right) \to {a_1}{x_1} \ldots \ldots + {a_n}{x_n}\)

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**Q.1. Find out the value of \({4^2} – {2^2}\)Ans:** We have:

\({4^2} – {2^2}\)

Using the formula \({a^2} – {b^2} = (a – b)(a + b)\)

Where \(a=4\) and \(b=2\)

\((a – b)(a + b)\)

\( = (4 – 2)(4 + 2)\)

\(=12\)

Hence, the required answer is \(12\)

**Q.2. \({4^3} \times {4^2} = ?\)Ans:** We have:

\({4^3} \times {4^2} = ?\)

Using the exponential formula \(\left( {{a^m}} \right)\left( {{a^n}} \right) = {a^{m + n}}\)

Where \(a=4\)

\({4^3} \times {4^2}\)

\( = {4^{3 + 2}}\)

\( = {4^5} = 1024\)

Hence, the required answer is \(1024.\)

**Q.3. Use algebra formula to find \({(2x – 3y)^2}\)Ans: **We have, \({(2x – 3y)^2}\)

We use the identity here \({(a – b)^2} = {a^2} – 2ab + {b^2}\) to expand this.

Here, \(a=2x\) and \(b=3y\)

We get \({(2x – 3y)^2} = {(2x)^2} – 2(2x)(3y) + {(3y)^2} = 4{x^2} – 12xy + 9{y^2}\)

Hence, the required answer is \({(2x – 3y)^2} = 4{x^2} – 12xy + 9{y^2}\)

**Q.4. By using the algebra formulas – identities, evaluate \(297 \times 303\)Ans: ** We have \(297 \times 303\)

The above product can be written as \((300 – 3) \times (300 + 3)\)

Now, we have to find the product by using the formula: \((a – b)(a + b) = {a^2} – {b^2}\)

Here, \(a=300\) and \(b=3.\)

We get \((300 – 3) \times (300 + 3) = {300^2} – {3^2} = 90000 – 9 = 89991\)

Hence, the required answer is \(297 \times \times 303 = 89991.\)

**Q.5. Find the roots of the quadratic equation |({x^2} + 5x + 6 = 0\) using algebra formulas for quadratic equations.Ans:** We have \({x^2} + 5x + 6 = 0\)

By comparing this with \(a{x^2} + bx + c = 0\)

We get \(a = 1;b = 5;c = 6\)

Substitute these values in quadratic formula:

\(x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}\)

\(x = \frac{{ – 5 \pm \sqrt {{5^2} – 4 \cdot 1 \cdot 6} }}{{2 \cdot 1}}\)

\(x = \frac{{ – 5 \pm \sqrt 1 }}{{2 \cdot 1}}\)

\(x = \frac{{ – 5 \pm 1}}{2},x = \frac{{ – 5 – 1}}{2}\)

\(x = – 2;x = – 3\)

Hence, \(x = – 2\) and \(-3\)

**Learn All the Concepts on Algebraic Identities**

Students might be having many questions regarding this topic. Here are a few commonly asked questions and answers.

**Q.1. What is the general formula in algebra?Ans:** The general algebra formulas that are used are given below:

\({(a + b)^2} = {a^2} + 2ab + {b^2}\)

\({(a – b)^2} = {a^2} – 2ab + {b^2}\)

\((a + b)(a – b) = {a^2} – {b^2}\)

\((x + a)(x + b) = {x^2} + x(a + b) + ab\)

**Q.2. What are the basics of algebra?Ans:** The basics of algebra consist of numbers, variables, constants, expressions, equations, linear equations, and quadratic equations. It also includes the basic arithmetic operations of addition, subtraction, multiplication, and division within the algebraic expressions.

**Q.3. What is the algebraic formula of \(AB\)?Ans:** \(AB\) is the most commonly used variables or letter that represent algebraic equations or problems.

\({(a + b + c)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2ac + 2bc\)

\({(a + b + c)^3} = {a^3} + {b^3} + {c^3} + 3(a + b)(b + c)(c + a)\)

**Q.4. What is an algebraic formula?Ans:** An algebraic formula is an equation and the rule written using the mathematical and the algebraic symbols. It involves algebraic expressions on both sides. The algebraic formula is the short quick formula to solve complex algebraic calculations.

**Q.5. What is the formula of algebraic identities?Ans:** The algebraic identities are as given below:

\({(a + b)^2} = {a^2} + 2ab + {b^2}\)

\({(a – b)^2} = {a^2} – 2ab + {b^2}\)

\({a^2} – {b^2} = (a + b)(a – b)\)

\((x + a)(x + b) = {x^2} + (a + b)x + ab\)