While it’s not obvious, learning equations are important to our daily lives. Without it, we would not have computers, GPS, satellite TV, and other inventions that make modern society what it is.
Some of the equations that changed the way we live are The Pythagorean Theorem, The fundamental theorem of calculus, and Newton’s universal law of gravitation, just to name a few.
The equation is essential to medicine, economics, computer science, engineering, and many more. Read on if you want to learn more about this dynamic concept.
Types of Equations – Algebraic
Cubic Equation
A cubic equation is a polynomial equation whereby the highest sum of exponents of the variables in any term is equal to three. In other words, it is an equation involving a cubic polynomial; i.e., one of the forms. It has the following form:
ax3 + bx2 + cx + d = 0 where a ≠ 0
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Exponential Equation
Exponential equations have variables in the place of exponents, and can be solved using this property: axax = ayay => x = y. Examples include the following:
- 4x = 0
- 8x = 32
- ab = 0 (where “a” is base and “b” is the exponent)
Irrational Polynomial Equation
Irrational polynomial equations are those equations with at least a polynomial under the radical sign.
Linear Equation
Linear equations are those where each term is either a constant or the product of a single variable and a constant. If there are two variables, the graph of the linear equation is always going to be a straight line. As a general rule, a linear equation looks like this:
y = mx + c, m ≠≠ 0
In this example, m is known as slope and c represents that point on which it cut the y axis.
In linear equations with different variables:
The equation with only one variable: an equation that has only one variable. Examples include the following:
- 8a – 8 = 0
- 9a = 72
The equation has two variables: an equation that has only two types of variables. Examples include the following:
- 9a + 6b – 82 = 0
- 7x + 7y = 12
- 8a – 8d = 74
The equation that has three variables: this is an equation with only three types of variables in the equation. Examples include the following:
- 13a – 8b + 31c = 74
- 5x + 7y – 6z = 12
- 6p + 14q – 74 + 82 = 0
Logarithmic Equation
These are equations whereby the unknown is always affected by a logarithm.
Polynomial Equation
Polynomial equations contain either variables or indeterminates and coefficients. These are involved in operations such as addition, subtraction, multiplication, and non-negative integer exponents. Examples include the following:
- ax + by + c = 0ax + by + c = 0 with degree = 1 and two variables
- ax2 + bx + c = 0ax2 + bx = c = 0 with degree = 2 and one variable
- ax + b = 0 with degree = 1 and one variable
- axy + c = 0axy + c = 0 with degree = 2 and two variables
Quadratic Equation
A quadratic equation is a second-degree equation whereby one variable contains the variable that has an exponent of two. An example and the general form is shown below.
ax2 + bx + c = 0, a ≠≠ 0
Other examples include:
- 5a2 – 5a = 35
- 8x2 + 7x – 75 = 0
- 4y2 + 14y – 8 = 0
Quartic Equation
Quartic equations are equations of the fourth degree and an equation that equates a quartic polynomial to zero, using this form:
f(x) = ax4 + bx3 + cx2 + dx + e = 0 where a ≠ 0
The derivative of a quartic function is a cubic function.
Quintic Equation
A quintic equation is a polynomial equation in which five is the highest power of the variable. The formula used is:
ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0
Examples include the following:
- x^5 + x^3 + x
- y^5 + y^4 + y^3 + y^2 + y + 1
Radical Equation
Radical equations are those that have a maximum exponent on the variable that is 12 and which have more than one term. It can also be said that a radical equation is one whereby the variable is lying inside a radical symbol, usually in the form of a square root. Examples include the following:
- + 10 = 26
- + x – 1
Rational Equation
A rational equation involves rational expressions.
Transcendental Equations
Transcendental equations are equations that include transcendental functions. Exponential equations are examples of transcendental equations.
Trigonometric Equation
Put simply, trigonometric equations are those equations that involve trigonometric functions, usually of unknown angles such as cos B = ½.
An example of a trigonometric equation is found here:
x =
More samples of trigonometric equations can be found here.
Other examples of algebraic equations can be found here.
Types of Equations – Geometric
Figures – Formulas for Volume (V) and Surface Area (SA)
General Prism
- V = Bh = area of base x height
- SA = sum of the areas of the faces
Rectangular Prism
- V = lwh = length x width x height
- SA = 2lw + 2hw + 2lh
- = 2(length x width) + 2(height x width) + 2(length x height)
Right Circular Cone
- V = Bh = x area of base x height
- SA = B + C
- = area of base + ( x circumference of base x slant height)
Right Circular Cylinder
- V = Bh = area of base x height
- SA = 2B + Ch = (2 x area of base) + (circumference x height)
Sphere
- V = 3 = x x cube of radius
- SA = 4 2 = 4 x x square of radius
Square Pyramid
- V = Bh = x area of base x height
- SA = B + P
- = area of base + ( x perimeter of base x slant height)
Shapes – Formulas for Area (A) and Circumference (C)
Circle
A = 2 = x square of radius
C = 2 r = 2 x x radius
C = = x diameter
Parallelogram
A = bh = base x height
Rectangle
A = lw = length x width
Trapezoid
A = (b1 + b2)h = x sum of bases x height
Triangle
A = bh = x base x height
Equations of a Line
Standard form
Ax + By = 0 where A and B are not zero
Point-Slope form
y – y1 = m(x – x1) where m = slop, (x1, y1) = point on line
Slope-Intercept form
y = mx + b or y = b + mx where m = slope b = y – intercept