Question

How do you solve ${{x}^{2}}+2x+10=0$ using the quadratic formula?

Answer 1

Hint: We will see the standard quadratic equation. Then we will compare the given quadratic equation with the standard quadratic equation and obtain the corresponding coefficients. Then we will look at the quadratic formula to solve the quadratic equation. We will substitute the coefficients of the given quadratic equation in the quadratic formula and simplify it to obtain the solution of the given quadratic equation.

Complete step-by-step solution:
We know that the standard quadratic equation is $a{{x}^{2}}+bx+c=0$. The given quadratic equation is ${{x}^{2}}+2x+10=0$. Now, we will compare the standard quadratic equation with the given quadratic equation. Comparing the two quadratic equations, we obtain the corresponding coefficients as $a=1$, $b=2$ and $c=10$.
The quadratic formula to obtain the solution of a standard quadratic equation is given as
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Now, we will substitute the values $a=1$, $b=2$ and $c=10$ in the quadratic formula to obtain the solution of the given quadratic equation, in the following manner,
$x=\dfrac{-2\pm \sqrt{{{2}^{2}}-4\times 1\times 10}}{2\times 1}$
Simplifying the above equation, we get
$\begin{align}
  & x=\dfrac{-2\pm \sqrt{4-40}}{2} \\
 & \Rightarrow x=\dfrac{-2\pm \sqrt{-36}}{2} \\
 & \Rightarrow x=\dfrac{-2\pm 6i}{2} \\
 & \therefore x=-1\pm 3i \\
\end{align}$
Therefore, the solution of the given quadratic equation is $x=-1\pm 3i$.

Note: It is important that we know the quadratic formula to solve this type of question. Since the method of solving the quadratic equation is mentioned in the question, we have to use it. But there are other ways to solve the given quadratic equation. The other methods are factorization method and completing the square method. We can use these methods to verify the answer obtained by the quadratic formula. It is better to do the calculations explicitly so that we can avoid making any minor mistakes.