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,

• (a+b)² is an algebraic expression.
• a² +2ab+b² is an algebraic expression.
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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. 

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.

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.

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, .....ar^n-1.  The below formulas are helpful to find the nth term and the sum of the terms of the arithmetic, and geometric sequence.

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. 

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.

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 \( [\vec{a} \text{  } \vec{b} \text{  } \vec{c} ] = \vec{a} \cdot (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \cdot \vec{c} \).

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.

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. 

• Algebra
• Introduction to Algebra
• Variable Expression
• Variables, Constants, and Expressions
• Polynomials in one Variable
• Algebra Basic Formulas
• Exponents
• Logarithms
• Simple Equation Calculator
• Reciprocal Calculator
• System of Equations Solver

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)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a+b)³ =a³ +3a²b + 3ab² + 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.

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