Question

How do you factor ${x^2} - x + 18$?

Answer 1

Hint: The given equation is of the form of a quadratic equation. Here we need to find the roots of the equation. i.e. we need to find the values of the variable x. Compare the given quadratic equation with the general quadratic equation and substitute the values in the formula of finding the roots of the equation.
If $a{x^2} + bx + c = 0$ is an general quadratic equation, then its roots are found using the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.

Complete step-by-step answer:
We are given the quadratic equation
${x^2} - x + 18$
We are asked to find the factors of the given expression.
Let us take ${x^2} - x + 18 = 0$ …… (1)
If $a{x^2} + bx + c = 0$ is an general quadratic equation, then we find its roots using the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. …… (2)
On comparing with the general quadratic equation $a{x^2} + bx + c = 0$, we get,
$a = 1, $ $b = - 1,$ $c = 18$.
Substitute the values of $a,$ $b$ and $c$ in the formula of finding the roots of the equation given by the equation (2), we get,
$ \Rightarrow x = \dfrac{{ - ( - 1) \pm \sqrt {{{( - 1)}^2} - 4(1)(18)} }}{{2 \times 1}}$
Square the values inside the square root in the numerator of the fraction and then we obtain the solution as,
$ \Rightarrow x = \dfrac{{1 \pm \sqrt {1 - 72} }}{2}$
Now calculate the value under the square root in the fraction we obtain,
$ \Rightarrow x = \dfrac{{1 \pm \sqrt { - 71} }}{2}$
 This can also be written as,
$ \Rightarrow x = \dfrac{{1 \pm \sqrt { - 1} \sqrt {71} }}{2}$
We know from complex numbers, the value of the imaginary unit is, $i = \sqrt { - 1} $.
Hence we get,
$ \Rightarrow x = \dfrac{{1 \pm i\sqrt {71} }}{2}$
Hence we obtain the roots as ${x_1} = \dfrac{{1 + i\sqrt {71} }}{2}$ and ${x_2} = \dfrac{{1 - i\sqrt {71} }}{2}$
So, the factors for the quadratic equation when the roots are given is found using the formula,
$(x - {x_1})(x - {x_2})$
Substituting the values of ${x_1}$ and ${x_2}$ we get the required factors.
$ \Rightarrow \left( {x - \left( {\dfrac{{1 + i\sqrt {71} }}{2}} \right)} \right)\left( {x - \left( {\dfrac{{1 - i\sqrt {71} }}{2}} \right)} \right)$
Hence the factors of the expression ${x^2} - x + 18$ is given by $\left( {x - \left( {\dfrac{{1 + i\sqrt {71} }}{2}} \right)} \right)\left( {x - \left( {\dfrac{{1 - i\sqrt {71} }}{2}} \right)} \right)$.

Note:
Here we can’t use the method of factorization to find the roots or values of x for this question. We need to keep in mind that the coefficient of x should be broken in such a way that its sum gives us the value of the coefficient x and its product gives the value of the coefficient of ${x^2}$.
It is also important to remember the formula to find the roots of the quadratic equation
$a{x^2} + bx + c = 0$ which is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, where $\Delta = \sqrt {{b^2} - 4ac} $ is the discriminant of the quadratic equation. The value of discriminant can be positive, negative or zero. And it tells us the nature of the roots.