Question

Check whether the following is a quadratic equation. ${{x}^{3}}-4{{x}^{2}}-x+1={{(x-2)}^{3}}$.

Answer 1

Hint: An equation in which the highest exponent of the variable is a square. For example, ${{x}^{2}},{{a}^{2}},{{c}^{2}}$ etc. generally quadratic equation is written as $a{{x}^{2}}+bx+c=0$. Now, we will find whether the given equation is quadratic or not.

Complete step-by-step answer:
It is given in the question that we have to check whether the equation ${{x}^{3}}-4{{x}^{2}}-x+1={{(x-2)}^{3}}$ is quadratic or not. As we know, in quadratic equations the maximum power of a variable is $2$. For example,
${{(x+2)}^{2}}={{x}^{2}}+4+2(2)(x)$
$={{x}^{2}}+4+4x$
Thus, in the above equation variable $x$ has the maximum power of $2$. Therefore, it is a quadratic equation. In general, the quadratic equation is in the form of $a{{x}^{2}}+bx+c=0$. Where, $a,b,c$ are constants and $a\ne 0$.
The given equation is, ${{x}^{3}}-4{{x}^{2}}-x+1={{(x-2)}^{3}}$ ……………………………….$(1)$
Initially the equation is given in cubic form, we will solve it to find whether it is a quadratic equation or not. We know that, ${{(a-b)}^{3}}={{a}^{3}}-{{b}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}$ from this formula we can expand ${{(x-2)}^{3}}={{(x)}^{3}}-{{(2)}^{3}}-3{{(x)}^{2}}(2)+3(x){{(2)}^{2}}$.
$\Rightarrow {{(x-2)}^{3}}={{x}^{3}}-8-6{{x}^{2}}+12x$
On arranging the above equation, we get
$\Rightarrow {{(x-2)}^{3}}={{x}^{3}}-6{{x}^{2}}+12x-8$ …………………………………$(2)$
Substituting equation $(2)$ in equation $(1)$,
$\Rightarrow {{x}^{3}}-4{{x}^{2}}-x+1={{x}^{3}}-8-6{{x}^{2}}+12x$
Cancelling the similar terms on both sides, we get
$\Rightarrow -4{{x}^{2}}+6{{x}^{2}}-x-12x+1+8=0$
$\Rightarrow 2{{x}^{2}}-13x+9=0$
Now, the equation obtained above is in the form of a quadratic equation as the variable $x$ has a maximum power of $2$. On comparing the obtained equation with general form of quadratic equation $a{{x}^{2}}+bx+c=0$, we get
$a=2,b=-13,c=9$.

Note: In general, the quadratic equation is in the form of $a{{x}^{2}}+bx+c=0$. Where, $a,b,c$ are constants and $a\ne 0$. There are few points that we must know about the nature of quadratic equations, they are –
1. The value of $D={{b}^{2}}-4ac$ is called the discriminant in the quadratic equation.
2. $x=\dfrac{-b\pm \sqrt{D}}{2a}$ is a general formula to find the roots of a quadratic equation.
4. If $D>0$ the roots of the quadratic equation are real but unequal.
5. If $D<0$ the roots of the quadratic equation are complex with non-zero and imaginary parts.