How do you solve ${x^2} - 6x + 6 = 0$ using the quadratic formula?

Hint: In this question, we are given a quadratic equation and we have been asked to solve it using the quadratic formula. Compare the equation with the standard quadratic equation $a{x^2} + bx + c = 0$ , write the value of each coefficient and put them in the quadratic formula. Simplify the formula and you will get your required answers.

Formula used: Quadratic Formula - $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Complete step-by-step solution:
We are given an equation ${x^2} - 6x + 6 = 0$. The standard quadratic equation is $a{x^2} + bx + c = 0$. On comparing both the equations, we will get –
$ \Rightarrow a = 1$, $b = - 6$ and $c = 6$.
Now, we will simply put the values in the formula.
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Putting the values,
$ \Rightarrow x = \dfrac{{ - \left( { - 6} \right) \pm \sqrt {{{\left( { - 6} \right)}^2} - 4 \times 1 \times 6} }}{{2 \times 1}}$
On simplifying the equation, we get,
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {36 - 24} }}{2}$
Let us subtract the value and we get,
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {12} }}{2}$
We can write $\sqrt {12} $ as $2\sqrt 3 $.
Putting in the formula,
$ \Rightarrow x = \dfrac{6}{2} \pm \dfrac{{2\sqrt 3 }}{2}$
Simplifying it further,
$ \Rightarrow x = 3 \pm \sqrt 3 $

Hence, our two final values are $3 + \sqrt 3 ,3 - \sqrt 3 $.

Note: 1) How to simplify $\sqrt {12} $? We will first find the prime factors of$12$.
$\begin{array}{*{20}{c}}
  {{\text{ }}2\left| \!{\underline {\,
  {12} \,}} \right. } \\
  {2\left| \!{\underline {\,
  6 \,}} \right. } \\
  {3\left| \!{\underline {\,
  3 \,}} \right. } \\
  1
\end{array}$
In this prime factorization, we can see a pair of$2$. So, we will take a $2$ out and write the other numbers inside the square root. Hence, $\sqrt {12} $ as $2\sqrt 3 $
2) We solved the given equation using the quadratic formula because it was asked in the question. If you try solving this question, using splitting the middle term, you will not be able to solve it. Why is it so?
${x^2} - 6x + 6 = 0$ 🡪 This is the given equation. We have to find two factors of $6$ such that they add up to $ - 6$. There are no such real factors of $6$, which will give us $ - 6$. So, we cannot solve using the middle term method. In such cases, the need to use the quadratic formula arises.