Question

How do you solve \[{x^2} + x + 10 = 0\] 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:
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], and by substituting the values given we will get the required values for \[x\].

Complete step-by-step answer:
Quadratic equations are the equations that are often called second degree. It means that it consists at least one term which is squared, the general form of quadratic equation is \[a{x^2} + bx + c = 0\], where "\[a\]", "\[b\]", and "\[c\]" are numerical coefficients or constant, and the value of\[x\]is unknown. And one fundamental rule is that the value of \[a\], the first constant cannot be zero in a quadratic equation.
Now the given quadratic equation is,
\[{x^2} + x + 10 = 0\],
Now using the quadratic formula, which is given by \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\],
Here, \[a = 1\],\[b = 1\],\[c = 10\],
Now substituting the values in the formula we get,
\[ \Rightarrow x = \dfrac{{ - \left( 1 \right) \pm \sqrt {{{\left( 1 \right)}^2} - 4\left( 1 \right)\left( {10} \right)} }}{{2\left( 1 \right)}}\],
Now simplifying we get,
\[ \Rightarrow x = \dfrac{{ - 1 \pm \sqrt {1 - \left( {40} \right)} }}{2}\],
Now again simplifying we get,
\[ \Rightarrow x = \dfrac{{ - 1 \pm \sqrt { - 39} }}{2}\],
As the negative number is under the square root so the roots are not real numbers and use the value of \[i\] i.e.,\[{i^2} = - 1\], we get,
Now simplifying we get,
\[ \Rightarrow x = \dfrac{{ - 1 \pm \sqrt {39 \times - 1} }}{2}\],
We know that\[{i^2} = - 1\], we get,
\[ \Rightarrow \]\[x = \dfrac{{ - 1 \pm \sqrt {39} i}}{2}\],
Now we get two values of \[x\] they are \[x = \dfrac{{ - 1 + \sqrt {39} i}}{2}\], \[x = \dfrac{{ - 1 - \sqrt {39} i}}{2}\].

\[\therefore \]If we solve the given equation, i.e.,\[{x^2} + x + 10 = 0\], then two values of \[x\] are \[x = \dfrac{{ - 1 + \sqrt {39} i}}{2}\],\[x = \dfrac{{ - 1 - \sqrt {39} i}}{2}\].

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 factorising 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.,\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].