A little revision on Quadratic Equations in the Class 8 section.

In elementary algebra, a quadratic equation is any equation having the form

ax^2+bx + c = 0

where x  represents an unknown, and a,\ b,\ and\ c  represent known numbers such that a \neq  0 .

If a = 0 , then the equation is linear, not quadratic.

The numbers a,\ b,\ and\ c  are the coefficients of the equation, and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.

The degree of a quadratic equation is 2.

Examples of quadratic equations:

21x^2-8x-4=0 \ where \ a=21, \ b=-8 \ and \ c=-4  

6x^2+5x-6=0 \ where \ a=6, \ b=5 \ and \ c=-6

Roots of Quadratic equations

Every quadratic equation ax^2+bx + c = 0  is satisfied by two values say p \  and \  q . These values, p \  and \  q , are said to be the root of the equation.

What this also means is that ax^2+bx + c  =  (x - p)(x - q) = 0

Note: Zero Product Rule: Whenever the product of the two expression is zero, then at least one of those is zero. In the above example;

(x-p)=0 \ or\  x=p

or

(x-q)=0 \ or\  x=q

Solving Quadratic equations

There are two ways to solve the quadratic equations.

Factorization Method

  • Step 1: Factorize ax^2+bx + c  =  (x - p)(x - q) = 0
  • Step 2: Equate each linear part to zero.
  • Step 3: Hence x = p \ and \ x = q

Using the Formula

  • Step 1: From the quadratic equation, first identify a,\ b,\ and\ c .
  • Then use the following formula

x =\frac{-b\pm \sqrt{b^2-4ac}}{2a}

Discriminant:

For a quadratic equation ax^2+bx + c = 0 where a \neq  0 ; expression b^2-4ac is called discriminant and is generally denoted by D . Thus, discriminant D = b^2-4ac .

Nature of the roots of the equations:

If a,\ b,\ and\ c  represent real numbers and a \neq  0 , then discriminant:

b^2-4ac=0 \Rightarrow the roots are real and equal.

b^2-4ac>0 \Rightarrow the roots are real and unequal.

b^2-4ac<0 \Rightarrow the roots are imaginary.

Square root of a negative number like \sqrt{-2},  \sqrt{-5}, \sqrt{-23}, \sqrt{-10} .... is an imaginary number.

Therefore, just be looking at D , we can tell the nature of the roots.

Note: Problems based on Geometrical Figures: The problems where you have a right triangle (where Pythagoras theorem is used) would see the application of the quadratic equation.

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