Question

How do you simplify the quadratic formula?

Answer 1

Hint: The quadratic formula is used for obtaining the solutions of a quadratic equation, or the roots of a quadratic polynomial. It is given by \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\], where $a$ is the coefficient of ${{x}^{2}}$, $b$ is the coefficient of $x$, and $c$ is the constant term of a quadratic equation. Or equivalently, it gives the solution of the quadratic equation $a{{x}^{2}}+bx+c=0$. The quadratic formula given above is already completely simplified algebraically. So we have to consider some special cases for some particular values of the coefficients.

Complete step by step answer:
We know that the quadratic formula is used for obtaining the solutions of a quadratic equation, or the roots of a quadratic polynomial. It is given by \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\], where $a$ is the coefficient of ${{x}^{2}}$, $b$ is the coefficient of $x$, and $c$ is the constant term of a quadratic equation $a{{x}^{2}}+bx+c=0$.
Now, the formula \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] is already simplified algebraically, so no algebraic operation can be performed for the simplification. But we can consider some of the special conditions for the values of the coefficients $a,b$ and $c$.
(i) The coefficient $b$ is an even number:
We know that every even number can be written in the form $b=2n$, where $n$ is some number. Therefore substituting $b=2n$ in the quadratic formula we get
\[\begin{align}
  & \Rightarrow x=\dfrac{-2n\pm \sqrt{{{\left( 2n \right)}^{2}}-4ac}}{2a} \\
 & \Rightarrow x=\dfrac{-2n\pm \sqrt{4{{n}^{2}}-4ac}}{2a} \\
 & \Rightarrow x=\dfrac{-2n\pm 2\sqrt{{{n}^{2}}-ac}}{2a} \\
 & \Rightarrow x=\dfrac{-n\pm \sqrt{{{n}^{2}}-ac}}{a} \\
\end{align}\]
Further, if the coefficient $a=1$ then in more simplified manner we can write the quadratic formula as
\[\Rightarrow x=-n\pm \sqrt{{{n}^{2}}-c}\]
Hence, this the required simplified form of the quadratic formula.

Note:
We can also simplify the quadratic formula by considering the case of repeated and imaginary roots. But those simplified versions of the formula will belong to the category of the particular cases only. So we have not considered these in the above solution.