Question

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

Answer 1

Hint: Take -3 common from all the terms and divide both the sides with it to simplify the equation. Now, write the constant term in its exponential form such that its exponent becomes 2. To do this, take the square root of the constant term and to balance it take its exponent equal to 2. Now, take the constant term to the R.H.S and take the square root on both the sides to get the two values of x and the answer.

Complete step by step answer:
Here, we have been provided with the quadratic expression: - \[-3{{x}^{2}}-9=0\] and we are asked to solve it. That means we have to find the values of x.
Now, as we can see that the given equation is quadratic in nature, so we must have two roots or values of x. Since, the coefficient of x in the above equation is 0, so we will not apply the middle term split method.
Clearly we can see that -3 is common in both the terms, so taking -3 common we get,
\[\Rightarrow -3\left( {{x}^{2}}+3 \right)=0\]
Dividing both the sides with -3 we get,
\[\Rightarrow {{x}^{2}}+3=0\]
Now, we can write the constant term 3 in its exponential form with the exponent equal to 2 as $3={{\left( \sqrt{3} \right)}^{2}}$, so we get,
\[\Rightarrow {{x}^{2}}+{{\left( \sqrt{3} \right)}^{2}}=0\]
Taking the constant term to the R.H.S we get,
\[\Rightarrow {{x}^{2}}=-{{\left( \sqrt{3} \right)}^{2}}\]
As we can see that the value of \[{{x}^{2}}\] is negative, that means we will not get any real roots of this equation. So, we need to find the complex roots,
Taking square roots both the sides we get,
\[\begin{align}
  & \Rightarrow x=\pm \sqrt{-{{\left( \sqrt{3} \right)}^{2}}} \\
 & \Rightarrow x=\pm \sqrt{-1}\times \sqrt{{{\left( \sqrt{3} \right)}^{2}}} \\
 & \Rightarrow x=\pm i\times \sqrt{3} \\
 & \Rightarrow x=\pm \sqrt{3}i \\
\end{align}\]

Hence, the roots of the given quadratic equation are: - $\sqrt{3}i$ and $-\sqrt{3}i$.

Note: One may also solve this equation by the discriminant method to get the answer. 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.