Question

Solve for x:
 ${{\left( {{x}^{2}}-x \right)}^{2}}-\left( {{x}^{2}}-x \right)-12=0$

Answer 1

Hint: Treat is as a quadratic equation and solve using middle term splitting. Then try to find x using the roots of the quadratic equation.

Complete step by step answer:
For our convenience, we let ${{x}^{2}}-x$ to be ‘t’.
Our equation becomes:
${{\left( {{x}^{2}}-x \right)}^{2}}-\left( {{x}^{2}}-x \right)-12=0$
$\Rightarrow {{t}^{2}}-t-12=0$
For a quadratic equation of the form $a{{x}^{2}}+bx+c=0$ , according to the rule of middle term splitting, we must try to find a pair of numbers such:
$\text{product of the numbers=ac}$
$\text{sum of the number=b}$
Put all the values of a, b and c with the respective signs.
Applying to our quadratic equation, we get:
${{t}^{2}}-t-12=0$
$\Rightarrow {{t}^{2}}-(4-3)t-12=0$
$\Rightarrow {{t}^{2}}-4t+3t-12=0$
$\Rightarrow t(t-4)+3(t-4)=0$
$\Rightarrow (t+3)(t-4)=0$
$\therefore t=4\text{ and -3}$
Now, we need to solve for each value of t.
First, taking t=4.
$\therefore {{x}^{2}}-x=4$
$\Rightarrow {{x}^{2}}-x-4=0$
A quadratic equation in x.
We know, for a quadratic equation of the form $a{{x}^{2}}+bx+c=0$ .
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Applying the formula to our quadratic equation, we have;
$x=\dfrac{-(-1)\pm \sqrt{{{(-1)}^{2}}-4\times 1\times (-4)}}{2\times 1}$
$\Rightarrow x=\dfrac{1\pm \sqrt{1+16}}{2}$
$\therefore x=\dfrac{1\pm \sqrt{17}}{2}$
Similarly, taking t=-3.
$\therefore {{x}^{2}}-x=-3$
$\Rightarrow {{x}^{2}}-x+3=0$
A quadratic equation in x.
Applying the formula of roots of a quadratic equation to our quadratic equation:
$\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
$\Rightarrow x=\dfrac{-(-1)\pm \sqrt{{{(-1)}^{2}}-4\times 1\times (3)}}{2\times 1}$
$\Rightarrow x=\dfrac{1\pm \sqrt{1-12}}{2}$
$\therefore x=\dfrac{1\pm \sqrt{-11}}{2}=\dfrac{1\pm i\sqrt{11}}{2}$
Therefore, all the values of x are: $\dfrac{1+\sqrt{17}}{2},\text{ }\dfrac{1-\sqrt{17}}{2},\text{ }\dfrac{1+i\sqrt{11}}{2},\text{ }\dfrac{1-i\sqrt{11}}{2}$ .

Note: Among the all four possible values of x mentioned above, we can quickly figure out that two are irrational and two are imaginary. If you are asked for real roots, report only the irrational roots. Also, be careful what the question asks: real roots, imaginary roots or rational roots and report the answer accordingly. As in the above question, x has 0 rational roots.
Point to remember is that Middle term splitting is mostly useful when you have rational roots in case of imaginary or irrational roots; it becomes difficult to figure out the numbers manually.