Question

How do you find the nature of the roots using the discriminant for \[9{x^2} + 6x + 1 = 0\] ?

Answer 1

Hint: Here the given equation is a quadratic equation. We will solve the given equation by using the quadratic formula and then find the roots. So to solve $ x $ we have: $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ . Here the term $ {b^2} - 4ac $ is called the discriminant. So to find the nature of the roots using the discriminant we need to substitute the necessary terms and find the discriminant.

Formula Used: $a{x^2} + bx + c = 0$ Here $a,\;b,\;c$ are numerical coefficients.

Complete step by step solution:
Given
\[9{x^2} + 6x + 1 = 0........................................\left( i \right)\]
Here we need to find the nature of the roots using the discriminant.
The discriminant can be found using the formula $ {b^2} - 4ac $ .
Also to find the nature of roots using the discriminant we have certain rules, which are:
If the discriminant is:
Positive: We will get two real solutions.
Negative: We will get complex solutions.
Zero: We will get only one solution
Now we need to compare (i) to the general formula and find the values of unknowns.
So on comparing (i) to the general formula $ a{x^2} + bx + c = 0 $ , we get:
 $ a = 9,\;b = 6,\;c = 1.....................\left( {ii} \right) $
On substituting it in the formula we can write:
 $
  {b^2} - 4ac = {\left( 6 \right)^2} - 4\left( 9 \right)\left( 1 \right) \\
   = 36 - 36 \\
   = 0.................................\left( {iii} \right) \\
  $
Therefore we got the discriminant is zero.
Now if the discriminant is zero then only one solution is possible.

Therefore for \[9{x^2} + 6x + 1 = 0\] there would be only one solution since the discriminant $ {b^2} - 4ac $ is zero.

Note: Quadratic formula is mainly used in conditions where grouping method cannot be used or when the polynomial cannot be reduced into some general identity.Quadratic formula method is an easier and direct method in comparison to other methods. Also while using the Quadratic formula when $ \sqrt {{b^2} - 4ac} $ is a negative root then the corresponding answer would be a complex number.