Question

How do you find the roots, real and imaginary of $y = {x^2} - 12x - 1$ using the quadratic formula?

Answer 1

Hint: In this question we have to solve the given polynomial by using quadratic formula, polynomial is in form $a{x^2} + bx + c$, where "$a$", "$b$", and “$c$" are real numbers and the Quadratic Formula is derived from the process of completing the square, and is formally stated as:
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, now substituting the values in the formula we will get the required value.

Complete step by step solution:
Now the given quadratic equation is,
$y = {x^2} - 12x - 1$,
Put y=0
Now using the quadratic formula, which is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$,
Here $a = 1$, $b = - 12$, $c = - 1$
Now substituting the values in the formula we get,
$ \Rightarrow x = \dfrac{{ - \left( { - 12} \right) \pm \sqrt {{{\left( { - 12} \right)}^2} - 4\left( 1 \right)\left( { - 1} \right)} }}{{2\left( 1 \right)}}$,
Now simplifying we get,
$ \Rightarrow x = \dfrac{{12 \pm \sqrt {144 - \left( { - 4} \right)} }}{2}$,
Now again simplifying we get,
$ \Rightarrow x = \dfrac{{12 \pm \sqrt {144 + 4} }}{2}$,
Further simplification we get,
$ \Rightarrow x = \dfrac{{12 \pm \sqrt {148} }}{2}$,
Now taking the square root we get,
$ \Rightarrow x = \dfrac{{12 \pm 2\sqrt {37} }}{2}$,
Now taking out the common terms we get,
$ \Rightarrow x = 6 \pm \sqrt {37} $,
So, the values of $x$ will be $6 \pm \sqrt {37} $

$\therefore $ If we solve the given equation, i.e., $y = {x^2} - 12x - 1$, then the values of $x$ are $6 + \sqrt {37} $ and $6 - \sqrt {37} $.

Note: Quadratic equation formula is a method of solving quadratic equations, but we should keep in mind that we can also solve the equation using completely the square, and we can cross check the values of $x$ by using the above formula. Also we should always convert the coefficient of ${x^2} = 1$, to easily solve the equation by this method, and there are other methods to solve such kind of solutions, other method used to solve the quadratic equation is by factoring method, in this method we should obtain the solution factorising quadratic equation terms. In these type of questions, we can solve by using quadratic formula i.e.,$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.