Quadratic Formula Derivation

Category: Algebra

"Published in Newark, California, USA"

Given the quadratic equation in general form:

                          
Prove that the Quadratic Formula has a formula of 

Solution:

There are three ways in solving the quadratic equation. First,  you can solve the quadratic equation by factoring. Second, if you cannot factor the quadratic equation, you can solve it by completing the square. Third, you can solve the quadratic equation by using the Quadratic Formula. Right now, we will derive the formula for Quadratic Formula. Let's consider the quadratic equation in general form

Divide both sides of the equation by the coefficient of x² which is a. 

Transpose the third term to the right side of the equation

Apply the completing the square method to the above equation

Get the LCD at the right side of the equation

Take the square root on both sides of the equation

Therefore,

Homogeneous Functions - Arbitrary Constant

Category: Differential Equations, Integral Calculus

"Published in Newark, California, USA"

Find the particular solution for 

when 

               

Solution:

If you examine the given equation, it is a differential equation because it has dy and dx in the equation. The type of equation is Homogeneous because the functions and variables cannot be separated by Separation of Variables. There's a method to solve the Homogeneous Functions. Consider the given equation

Let y = vx

    dy = vdx + xdv

Substitute y and dy to the above equation, we have

The above equation can now be separated by Separation of Variables. Arrange the equation according to their variables

Integrate both sides of the equation

Take the inverse natural logarithm on both sides of the equation

but y = vx and v = ^y/_x, 

To solve for C, substitute the following:

to the above equation, we have

Therefore,

  

Dividing Rational Fractions

Category: Algebra

"Published in Suisun City, California, USA"

Perform the indicated operations and simplify

Solution:

This is a division of a rational fraction with another rational fraction. As a rule in Mathematics that we need to get the reciprocal of the divisor first and then perform the multiplication as follows

Factor all the polynomials in the numerator and denominator. Do this by trial and error so that the middle term of a polynomial is matched.

Simplify the above equation

Therefore, the answer is