Quadratic Equations

       
           

Quadratic Equations are univariate polynomial equations of second degree and the standard form of Quadratic equation is shown below-

p x2 + q x + c = 0,

Here p, q and r are the constants with the condition that ‘p’ is not equal to zero. The constants p, q and r are named as quadratic coefficients, linear coefficient and constant term respectively.

As this is the second degree polynomial the solution of this expression have two roots and may or may not be distinct. This is also not necessary that the roots will be real the roots can have complex value. The solutions of the maths quadratic equations are given where the equation becomes zero. These two solutions are named as “roots” and “zeros”.

The roots of the quadratic equation solution are given as

x = - q + √ (q2 – 4 pr) / 2 p,

x = - q - √ (q2 – 4 pr) / 2 p,

The roots of the quadratic expression can be found as following-

p x2 + q x + c = 0,

On completing the square

x2 + (q / p) x = - r / p,

(x + (q / 2 p))2 = - r / p + q2 / 4 p2 = (q2 – 4 p r) / 4 p2,

x + (q / 2 p) = + √ (q2 – 4 pr) / 2 p and - √ (q2 – 4 pr) / 2 p,

On solving the expression for the variable ‘x’ we get

x = - q + √ (q2 – 4 pr) / 2 p,

x = - q - √ (q2 – 4 pr) / 2 p,

The name quadratic comes from the word “quad” that have the meaning square because the first term in the expression gets squared.

The term (q2 – 4 pr) is called as discriminant and there are three facts regarding this term are-

  • The discriminant is positive if there are two real solutions.
  • The discriminant is zero if there is only one real solution of the equation.
  • The Discriminant is negative if the equation has complex solutions.

Some higher degree equations are solved after their conversion in the quadratic equation and after getting the roots of the converted form the higher degree equations are solved. This is the application of quadratic equation.

The Quadratic Formula

An equation whose highest degree is equals to 2 is known as quadratic equation. In other words we can say that equation whose highest power is a square is said to be a quadratic equation. Quadratic equation can be written as: ax2 + bx + c = 0 and Quadratic Formula is given by:

⇨ x = - b + √ (b2 – 4ac) / 2a, its alternate form is also given by:

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Using the Square Root Property and Factoring

Square root of any number 'P' is a number 'Q' such that Q2 = P. Then 'Q' will be obtained by the expression Q = √P. Symbol '√' is known as radical symbol or root symbol. Cube root or fourth root or nth root will be shown as:

Cube root is 3√y, fourth root will be 4√y and 100th root will be written as 100√y.


Let’s discuss ...Read More

Applications of Quadratic Equations

Quadratic equation is a two degree polynomial equation, since highest degree is two it is also known as second degree equation. Degree is defined for variables, not for constants. We can find Applications of Quadratic Equations in various fields of mathematics.
 
General form of a quadratic equation is:
 
a y2 + b y + c = 0,
 
Maximum power is 2 f...Read More

Completing the Square

Multiplication of a number by itself is called square. x2 + 4x + 4 is a quadratic equation which is square of (x + 2)2. So we can say some quadratic equations are very simple to solve because they are in form of something with 'x' squared equals to some number and to solve these type of equations we will take square root of both sides, (x - 1)2 = 9, to s...Read More

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