Question

How do you solve \[9{{x}^{2}}+12x+4=0\] using the quadratic formula?

Answer 1

Hint: Compare the given quadratic equation with the general form given as \[a{{x}^{2}}+bx+c=0\]. Find the respective values of a, b and c. Now, find the discriminant of the given quadratic equation by using the formula: - \[D={{b}^{2}}-4ac\], where ‘D’ = discriminant. Now, apply the formula: - \[x=\dfrac{-b\pm \sqrt{D}}{2a}\] and substitute the required values to get the answer.

Complete step by step solution:
Here, we have been provided with the quadratic equation: - \[9{{x}^{2}}+12x+4=0\] and we are asked to solve it. That means we have to find the values of x. So, let us apply the discriminant method to solve the given quadratic equation.
Now, comparing the general form of a quadratic equation given as \[a{{x}^{2}}+bx+c=0\] with the given equation \[9{{x}^{2}}+12x+4=0\], we can conclude that we have,
\[\Rightarrow \] a = 9, b = 12 and c = 4
Applying the formula for discriminant of a quadratic equation given as: - \[D={{b}^{2}}-4ac\], where ‘D’ = discriminant, we get,
\[\begin{align}
  & \Rightarrow D={{12}^{2}}-4\left( 9 \right)\left( 4 \right) \\
 & \Rightarrow D=144-144 \\
 & \Rightarrow D=0 \\
\end{align}\]
Now, we know that the solution of a quadratic equation in terms of its discriminant value is given as: -
\[\Rightarrow x=\dfrac{-b\pm \sqrt{D}}{2a}\]
So, substituting the given values and obtained value of D, we get,
\[\begin{align}
  & \Rightarrow x=\dfrac{-12\pm \sqrt{0}}{2\times 9} \\
 & \Rightarrow x=\dfrac{-12}{18} \\
 & \Rightarrow x=\dfrac{-2}{3} \\
\end{align}\]
Hence, the solution of the given quadratic equation is \[x=\dfrac{-2}{3}\].

Note: One may think that here we have obtained only one value of x as the solution of the quadratic equation. The reason is that the value of discriminant is 0. Actually, we have two solutions but both are the same, i.e., \[x=\dfrac{-2}{3}\]. Note that we can also solve the question by directly converting \[9{{x}^{2}}+12x+4=0\] into \[{{\left( 3x+2 \right)}^{2}}=0\] because it is of the form \[{{a}^{2}}+2ab+{{b}^{2}}\] whose factored form is given as: - \[{{a}^{2}}+2ab+{{b}^{2}}={{\left( a+b \right)}^{2}}\]. You can also apply the middle term split method to solve the question and check the answer.