Question

How do you solve \[3{{x}^{2}}+9x-6=0\]?

Answer 1

Hint: Divide both the sides of the given quadratic equation with 3 to simplify. 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’ is the notation for the 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 a quadratic equation: - \[3{{x}^{2}}+9x-6=0\] 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, here we can see that 3 is common in all the terms, so dividing both the sides with 3, we get the simplified quadratic equation as \[{{x}^{2}}+3x-2=0\],
Now, comparing the general form of a quadratic equation: - \[a{{x}^{2}}+bx+c=0\] with the given equation \[{{x}^{2}}+3x-2=0\], we can conclude that we have,
\[\Rightarrow \] a = 1, b = 3 and c = -2.
Applying the formula for discriminant of a quadratic equation given as: - \[D={{b}^{2}}-4ac\], where ‘D’ is the discriminant, we get,
\[\begin{align}
  & \Rightarrow D={{\left( 3 \right)}^{2}}-4\times 1\times \left( -2 \right) \\
 & \Rightarrow D=9+8 \\
 & \Rightarrow D=17 \\
\end{align}\]
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 values of D, we get,
\[\begin{align}
  & \Rightarrow x=\dfrac{-3\pm \sqrt{17}}{2\times 1} \\
 & \Rightarrow x=\dfrac{-3\pm \sqrt{17}}{2} \\
\end{align}\]
(i) Considering (+) sign we have,
\[\Rightarrow x=\dfrac{-3+\sqrt{17}}{2}\]
(ii) Considering (-) sign we have,
\[\Rightarrow x=\dfrac{-3-\sqrt{17}}{2}\]

Note: One may note that we will not be able to use the middle term split method to solve the question. This is because it will be difficult for us to think of the factors like: \[\left( x-\left( \dfrac{-3+\sqrt{17}}{2} \right) \right)\] and \[\left( x-\left( \dfrac{-3-\sqrt{17}}{2} \right) \right)\]. You may apply the third method known as completing the square method to solve the question. In this method we have to convert the equation \[a{{x}^{2}}+bx+c=0\] into the form: - \[{{\left( x+\dfrac{b}{2a} \right)}^{2}}=\dfrac{D}{4{{a}^{2}}}\] and then by taking the square root we need to solve for the value of x. Remember that the discriminant formula is derived from completing the square method.