Question

How do you solve ${x^2} - 4x + 13 = 0$ using quadratic formula ?

Answer 1

Hint: In this question we have to solve the given polynomial using quadratic formula. In the polynomial$a{x^2} + bx + c$, 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:
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Complete step by step answer:
Now the given quadratic equation is,
${x^2} - 4x + 13 = 0$,
Now using the quadratic formula, which is given by$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$,
Here$a = 1$,$b = - 4$, and $c = 13$,
Now substituting the values in the formula we get,
$ \Rightarrow x = \dfrac{{ - \left( { - 4} \right) \pm \sqrt {{{\left( { - 4} \right)}^2} - 4\left( 1 \right)\left( {13} \right)} }}{{2\left( 1 \right)}}$,
Now simplifying we get,
$ \Rightarrow x = \dfrac{{4 \pm \sqrt {16 - 52} }}{2}$,
Now again simplifying we get,
$ \Rightarrow x = \dfrac{{4 \pm \sqrt { - 36} }}{6}$,
As we know that square root of a negative number does not exist we will use the fact that ${i^2} = - 1$ , and substituting the value in the above we get,
Now taking the square root we get,
$ \Rightarrow x = \dfrac{{4 \pm \sqrt { - 1} \times \sqrt {36} }}{2}$,
Now we know that ${i^2} = - 1$, so simplifying we get,
$ \Rightarrow x = \dfrac{{4 \pm \left( {i \times 6} \right)}}{2}$,
Now taking common terms we get,
$ \Rightarrow x = 2 \pm 3i$
So the values of $x$ are $2 \pm 3i$.

$\therefore $If we solve the given equation, i.e., ${x^2} - 4x + 13 = 0$, then the value of $x$ are $2 \pm 3i$.

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 types of questions, we can solve by using quadratic formula i.e.,$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.