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.

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

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

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

$(x-q)=0 \ or\ x=q$

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.