Important Formulas: Algebra | SSC CGL Tier 2 - Study Material, Online Tests, Previous Year PDF Download

Basic Algebra Formulas

Addition: a + b
Adds two algebraic expressions

• Follows the commutative property i.e a + b = b + a
• Used to solve addition questions

Subtraction: a - b
Subtracts one algebraic expression from another

• Does not follow the commutative property
• Used to solve subtraction questions

Multiplication: a * b
Multiplies two algebraic expressions

• Follows both commutative and associative properties:
  a * b = b * a and a * (b * c) = (a * b) * c
• Used to solve multiplication questions

Division: a / b
Divides one algebraic expression by another

• Does not follow the commutative property
• The reciprocal property holds i.e (a/b) = a * (1/b)
• Used to solve division questions

Intermediate Algebra Formulas

Quadratic Formula: The quadratic formula is a formula used to solve quadratic equations. A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a is not equal to 0.
The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a

where:

• x is the solution to the quadratic equation
• a, b, and c are the coefficients of the quadratic equation
• √(x) is the square root of x

The quadratic formula can be used to solve any quadratic equation. To use the quadratic formula, you first need to identify the values of a, b, and c in the equation. Once you have identified these values, you can plug them into the quadratic formula and solve for x.
The quadratic formula can be used to solve quadratic equations in both real and complex numbers. However, if the discriminant (b² - 4ac) is negative, then the quadratic formula will produce two complex solutions.

Arithmetic Progression: Arithmetic progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant. The formula for the nth term of an AP is:
a_n = a + (n-1)d
where:

• a_n is the nth term
• a is the first term
• d is the common difference

For example, the sequence 1, 4, 7, 10, 13 is an arithmetic progression with first term 1 and common difference 3.
The formula for the sum of the first n terms of an AP is:
S_n = n/2 (2a + (n-1)d)
where:

• S_n is the sum of the first n terms
• n is the number of terms

For example, the sum of the first 5 terms of the arithmetic progression 1, 4, 7, 10, 13 is 35.
The arithmetic progression formulas can be used to solve a variety of problems involving arithmetic progressions. For example, you can use the formulas to find the nth term of an AP, the sum of the first n terms of an AP, or the average of the first n terms of an AP.

Geometric Progression: Geometric progression (GP) is a sequence of numbers where the ratio between any two consecutive terms is constant. The formula for the nth term of a GP is:
a_n = a r^(n-1)
where:

• a_n is the nth term
• a is the first term
• r is the common ratio

For example, the sequence 1, 2, 4, 8, 16 is a geometric progression with first term 1 and common ratio 2.
The formula for the sum of the first n terms of a GP is:
S_n = a(r^{n - 1})/(r-1)
where:

• S_n is the sum of the first n terms
• n is the number of terms

For example, the sum of the first 5 terms of the geometric progression 1, 2, 4, 8, 16 is 31.

The geometric progression formulas can be used to solve a variety of problems involving geometric progressions. For example, you can use the formulas to find the nth term of a GP, the sum of the first n terms of a GP, or the average of the first n terms of a GP.

Triangle: The area of a triangle is equal to half the product of its base and height. The formula for the area of a triangle is:
Area = (1/2) * base * height
where:

• Area is the area of the triangle
• base is the length of the base of the triangle
• height is the length of the height of the triangle

For example, the area of a triangle with a base of 10 cm and a height of 5 cm is 25 cm².

Circle: The area of a circle is equal to pi multiplied by the square of the radius. The formula for the area of a circle is:
Area = pi * r²
where:

• Area is the area of the circle
• pi is a mathematical constant approximately equal to 3.14
• r is the radius of the circle

For example, the area of a circle with a radius of 5 cm is 78.54 cm².

Cuboid: The volume of a cuboid is equal to the product of its length, width, and height. The formula for the volume of a cuboid is:
Volume = length * width * height
where:

• Volume is the volume of the cuboid
• length is the length of the cuboid
• width is the width of the cuboid
• height is the height of the cuboid

For example, the volume of a cuboid with a length of 10 cm, a width of 5 cm, and a height of 3 cm is 150 cm³.

Advanced Algebra Formulas

Binomial expansion: Binomial expansion is a formula that expands the powers of a binomial. A binomial is a sum of two terms, such as (a + b). The binomial expansion formula is:

(a + b)ⁿ = nC₀ a^{n + n}C₁ a^(n-1) b + nC₂ a^(n-2) b² + ... + nC_n-1 ab^(n-1) + bⁿ

where:

• n is the power of the binomial
• a and b are the terms of the binomial

nCk is the binomial coefficient, which is the number of ways to choose k objects from a set of n objects.
For example, the binomial expansion of (a + b)² is:
(a + b)² = 2a² + 2ab + b²
The binomial expansion formula can be used to expand the powers of any binomial. For example, you can use the formula to expand the powers of (a + b), (x + y), or (p + q).

Straight Line Equation: A linear equation is an equation of the form ax + by = c,  where a, b, and c are real numbers and a ≠ 0.
The line equation is also known as a slope-intercept form because it can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept.
The slope of a line is a measure of how steep the line is. The slope of a line can be calculated by dividing the change in y by the change in x. The y-intercept is the point where the line crosses the y-axis.
The straight line equation can be used to represent any straight line. For example, the equation y = 2x + 3 represents a line with a slope of 2 and a y-intercept of 3.
The straight line equation can be used to solve a variety of problems involving straight lines. For example, you can use the equation to find the slope or y-intercept of a line, or to find the point where two lines intersect.

Circle Equation: The equation of a circle is a mathematical equation that describes a circle. The standard equation of a circle with center (h, k) and radius r is:
(x - h)² + (y - k)² = r²
where:
x and y are the coordinates of a point on the circle
h and k are the coordinates of the center of the circle
r is the radius of the circle
The equation of a circle can be used to find the coordinates of any point on the circle, or to find the radius of the circle. For example, the equation (x - 2)² + (y - 3)² = 5² represents a circle with center (2, 3) and radius 5.
The equation of a circle can also be used to solve problems involving circles. For example, you can use the equation to find the distance between a point and a circle, or to find the area of a circle.

Area of a parallelogram: The area of a parallelogram is equal to the product of its base and height. The formula for the area of a parallelogram is:
Area = base * height
where:

• Area is the area of the parallelogram
• base is the length of the base of the parallelogram
• height is the length of the height of the parallelogram

For example, the area of a parallelogram with a base of 10 cm and a height of 5 cm is 50 cm².

Volume of a sphere: The volume of a sphere is equal to (4/3)πr³, where π is a mathematical constant approximately equal to 3.14 and r is the radius of the sphere.
Volume = (4/3)πr³
where:

• Volume is the volume of the sphere
• π is a mathematical constant approximately equal to 3.14
• r is the radius of the sphere

For example, the volume of a sphere with a radius of 5 cm is 523.6 cm³.

Volume of a cylinder: The volume of a cylinder is equal to πr²h, where π is a mathematical constant approximately equal to 3.14, r is the radius of the cylinder, and h is the height of the cylinder.
Volume = πr²h

where:

• Volume is the volume of the cylinder
• π is a mathematical constant approximately equal to 3.14
• r is the radius of the cylinder
• h is the height of the cylinder

For example, the volume of a cylinder with a radius of 5 cm and a height of 10 cm is 1570.8 cm³.

Perimeter of a rectangle: The perimeter of a rectangle is equal to the sum of all four sides of the rectangle. The formula for the perimeter of a rectangle is:
Perimeter = 2(l + b)
where:

• Perimeter is the perimeter of the rectangle
• l is the length of the rectangle
• b is the breadth of the rectangle

For example, the perimeter of a rectangle with a length of 10 cm and a breadth of 5 cm is 30 cm.

Pythagorean theorem: The Pythagorean theorem is a mathematical equation that relates the lengths of the sides of a right triangle. The theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
a² + b² = c²
where:

• a and b are the lengths of the other two sides of the triangle
• c is the length of the hypotenuse

For example, in a right triangle with legs of length 3 cm and 4 cm, the length of the hypotenuse is 5 cm.