Quadratic Equation in Grade IX

A Quadratic equation is a polynomial equation of second degree. In general form a quadratic equation is written as a x ² + b x + c. In this x is our variable and a, b and c are our constants. But we should keep one condition in mind that is a ≠ 0, because if a = 0 then our equation is no more a quadratic equation.

For example -

3 x ² + 4 x + 6,

9 x ² + 4,

8 x ² + 4 x.

A very important thing that we should keep in our mind is that a quadratic equation always has two roots or in a simple manner there are two values of x. The two solutions may or may not be distinct.

To find the value of the variable x in all the above equations, we can use the quadratic formula.

The two values of x can be written as,

We can even get complex values.

In the above formula the term b ² - 4ac is known as the discriminant. It is written as

if D = 0, then we have only one real root of the equation and it is

x= -b / 2a.

The other method of solving a quadratic equation is by splitting the middle term.

In this method we split the middle term i.e. the coefficient of x in such a manner that we easily get a common term and we can find the value of x. This method is also called as the factoring method.

The general equation is ax ² + b x + c. We follow some general steps to solve the quadratic equation.

1. Multiply the constants a and c without forgetting their signs.

2. Write down b, with its sign.

3. Write all the possible factors of the product of ac and find which pairs add up to give b .

4. Now rewrite the x terms as the sum of two terms with these selected numbers as the coefficients.

So by this we can find the solution of any quadratic equation.

For Example –

1. Solve x ² + 5 x + 6.

Answer – the possible combination of the middle term is 1 x + 4 x and 2 x + 3 x.

The product of a and c is 6 x 1= 6. So the second combination is the suitable option for this quadratic equation. So now it can be written as x ² + 3 x + 2 x + 6.

Now we take x common from the first two terms and 2 from the last two terms.

Now the quadratic equation becomes x ( x + 3 ) + 2 ( x + 3 ).

This time we take (x + 3) common, so the equation now becomes (x + 2 ) ( x+ 3 ).

The value of x = -3, -2.

The solution for the quadratic equation x ² + 5 x + 6 is x = -2, -3.

So now we have all the basic knowledge we need for solving problems based on quadratic equations.