Question

How do you solve ${{x}^{2}}-4x-1=0$ using the quadratic formula?

Answer 1

Hint: In this problem we need to solve the given quadratic equation i.e., we need to calculate the values of $x$ where the given equation is satisfied. For solving a quadratic equation, we have several methods. But in the problem, they have mentioned to use the quadratic formula which is given by $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. For this we need to compare the given equation with the standard quadratic equation $a{{x}^{2}}+bx+c=0$ and write the values of $a$, $b$, $c$. Now we will substitute those values in the formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ and simplify the obtained equation to get the required result.

Complete step by step answer:
Given equation ${{x}^{2}}-4x-1=0$.
Comparing the above quadratic equation with standard quadratic equation $a{{x}^{2}}+bx+c=0$, then we will get the values of $a$, $b$, $c$ as
$a=1$, $b=-4$, $c=-1$.
We have the quadratic formula for the solution as
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Substituting the values of $a$, $b$, $c$ in the above equation, then we will get
$\Rightarrow x=\dfrac{-\left( -4 \right)\pm \sqrt{{{\left( -4 \right)}^{2}}-4\left( 1 \right)\left( -1 \right)}}{2\left( 1 \right)}$
We know that when we multiplied a negative sign with the negative sign, then we will get a positive sign. Applying the above rule and simplifying the above equation, then we will get
$\begin{align}
  & \Rightarrow x=\dfrac{4\pm \sqrt{16+4}}{2} \\
 & \Rightarrow x=\dfrac{4\pm \sqrt{20}}{2} \\
\end{align}$
In the above equation we have the value $\sqrt{20}$. We need to simplify this value to get the simplified result. We can write $20=4\times 5={{2}^{2}}\times 5$, then the value of $\sqrt{20}$ will be $\sqrt{20}=\sqrt{{{2}^{2}}\times 5}=2\sqrt{5}$. Substituting this value in the above equation, then we will get
$\Rightarrow x=\dfrac{4\pm 2\sqrt{5}}{2}$
Taking $2$ as common in the numerator, then we will get
$\begin{align}
  & \Rightarrow x=\dfrac{2\left( 2\pm \sqrt{5} \right)}{2} \\
 & \Rightarrow x=2\pm \sqrt{5} \\
\end{align}$

Hence the solution of the given quadratic equation ${{x}^{2}}-4x-1=0$ is $2\pm \sqrt{5}$.

Note: In this problem they have mentioned to use the quadratic formula to solve the equation. If they have not mentioned the method then we can use any method for solving the equation. We have another method i.e., factorization method which will factorize the given equation and equate each factor to zero to get the solution.