Question

How do you solve ${x^2} - 6x - 3$ using quadratic formula?

Answer 1

Hint: In this question, we are given a quadratic equation and we have been asked to solve it. But it is also mentioned that we have to solve it using the quadratic formula.
Simply, compare the given equation with $a{x^2} + bx + c = 0$ , and write the values of the variables. Put these values in the formula and then simplify to get the answer.

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

Complete step-by-step solution:
We are given an equation ${x^2} - 6x - 3$ . 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 $c = - 3$.
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 \left( { - 3} \right)} }}{{2 \times 1}}$
On simplifying the equation, we get,
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {36 + 12} }}{2}$
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {48} }}{2}$
We can write $\sqrt {48} $ as $4\sqrt 3 $.
Putting in the formula,
$ \Rightarrow x = \dfrac{6}{2} \pm \dfrac{{4\sqrt 3 }}{2}$
Simplifying it further,
$ \Rightarrow x = 3 \pm 2\sqrt 3 $

Hence, our two final values are $3 + 2\sqrt 3 $ , $3 - 2\sqrt 3 $.

Note: 1) How to simplify $\sqrt {48} $? We will first find the prime factors of $48$.
$\begin{array}{*{20}{c}}
  {{\text{ }}2\left| \!{\underline {\,
  {48} \,}} \right. } \\
  {{\text{ }}2\left| \!{\underline {\,
  {24} \,}} \right. } \\
  {{\text{ }}2\left| \!{\underline {\,
  {12} \,}} \right. } \\
  {2\left| \!{\underline {\,
  6 \,}} \right. } \\
  {3\left| \!{\underline {\,
  3 \,}} \right. } \\
  1
\end{array}$
In this prime factorization, we can see two pairs of $2$. So, we will take a $2$ out of each pair, multiply them together and write the other product outside the square root and the unpaired numbers inside the square root. Hence, $\sqrt {48} $ as $4\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 method, you will not be able to solve it. Why is it so?
${x^2} - 6x - 3$ 🡪 This is the given equation. We have to find two factors of $ - 3$ such that they add up to $ - 6$ . There are no such real factors of $ - 3$, which will give us $ - 6$. So, we cannot solve using the middle term method.