Question

How do you find the roots of \[y={{x}^{2}}+3x+6\]?

Answer 1

Hint: The given question is algebraic expression which consists of variables, coefficients, constants and mathematical operations, such as addition, subtraction, multiplication and division. In the given question of an expression, we just need to simplify the expression by using the quadratic equation \[a{{x}^{2}}+bx+c=0\]. Roots of the quadratic equation = \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\].

Complete step by step solution:
The quadratic formula provides the solution for the quadratic equation:
 \[a{{x}^{2}}+bx+c=0\], in which a, b and c are the coefficients of respective terms in the quadratic equation, as follows:
Roots of the quadratic equation = \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\].
Given quadratic equation,
\[y={{x}^{2}}+3x+6\]
\[= {{x}^{2}}+3x+6=0\]
The quadratic formula provides the solution for the quadratic equation:
 \[a{{x}^{2}}+bx+c=0\], in which a, b and c are the coefficients of respective terms in the quadratic equation, as follows:
Roots of the quadratic equation = \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\].
Determine the quadratic equation’s coefficients a, b and c:
The coefficient of the given quadratic equation \[{{x}^{2}}+3x+6=0\]are,
\[\begin{align}
  & a=1 \\
 & b=3 \\
 & c=6 \\
\end{align}\]
Using these coefficient values into the quadratic formula:
 \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]\[=\dfrac{-3\pm \sqrt{{{\left( 3 \right)}^{2}}-\left( 4\times 1\times 6 \right)}}{2\left( 1 \right)}\]
Solve exponents and square root, we get
\[= \dfrac{-3\pm \sqrt{9-\left( 4\times 1\times 6 \right)}}{2\left( 1 \right)}\]
Performing any multiplication and division given in the formula,
\[\begin{align}
  & = \dfrac{-3\pm \sqrt{9-\left( 24 \right)}}{2} \\
 & = \dfrac{-3\pm \sqrt{-15}}{2} \\
\end{align}\]
Simplifying the above, we get
\[= \dfrac{-3\pm \sqrt{-15}}{2}\]
Here the value of discriminant i.e., \[d=\sqrt{{{b}^{2}}-4ac}=\sqrt{-15}\]is negative.
Therefore, no real solution because the discriminant is negative and the square root of a negative number is not a real number.
 This equation will have only imaginary roots and no real roots.
Now solving further,
\[= \dfrac{-3\pm \sqrt{-15}}{2}\]
We got two values i.e.
\[= \dfrac{-3+\sqrt{-15}}{2}\]and \[= \dfrac{-3-\sqrt{-15}}{2}\]
As we know that \[\sqrt{-1}=i\] which is an imaginary number
Thus, the imaginary roots will be:

\[x= \dfrac{-3-i\sqrt{15}}{2}and\dfrac{-3+i\sqrt{15}}{2}\]
Hence, this is the required answer.

Note:
To solve or evaluate these kinds of expressions, we need to know about solving quadratic equations using the formula, simplifying radicals and finding prime factors. In the roots of a quadratic equation formula, there is \[\pm \]students should remember this if they only write plus, they will only get one root.