Question

Which of the following is a quadratic equation?
A) \[{x^3} + 2x + 1 = {x^2} + 3\]
B) \[ - 2{x^2} = (5 - x)\left( {2x - \dfrac{2}{5}} \right)\]
C) \[(k + 1){x^2} + \dfrac{3}{2}x = 7\], where \[k = - 1\]
D) \[{x^3}{x^2} = {\left( x \right)^3}\]

Answer 1

Hint: To check if an equation is quadratic or not, always keep in mind: what is a quadratic equation?
Quadratic equation is one in which the highest power of the variable is two.
Remember general form of a quadratic equation is:
\[a{x^2} + {\text{ }}bx{\text{ }} + {\text{ }}c = 0\]
Where, \[a \ne 0\]. If \[a = 0\], it will become an equation in one variable.

Complete step-by-step answer:
In this question we will simplify all the four options one by one and check, in which option the highest power of the variable is two.
First we will start with option (a):
i.e. \[{x^3} + 2x + 1 = {x^2} + 3\]
On rearranging the equation, we get
$\Rightarrow$\[{x^3} + 2x + 1 = {x^2} + 3\]
Or \[{x^3}-{\text{ }}{x^2} + 2x + 1 - 3 = 0\]
It is clearly seen that the highest power of the variable is 3, so it is not a quadratic equation.
Now we will check for option (b)
i.e. \[ - 2{x^2} = (5 - x)\left( {2x - \dfrac{2}{5}} \right)\]
On rearranging the equation, we get
$\Rightarrow$\[ ({- 10 - \dfrac{2}{5}}) \times x +2 =0 \]
Clearly it can be seen that, highest power of a variable is one, hence it is also incorrect.
Now let us check for option (c):
\[(k + 1){x^2} + \dfrac{3}{2}x = 7\] where \[k = - 1\]
Substitute the value of k in this equation, we get-
$\Rightarrow$\[\left( {1 - 1} \right){x^2} + \dfrac{3}{2}x = 7\]
Or
$\Rightarrow$\[0{x^2} + \dfrac{3}{2}x = 7\]
Or \[\dfrac{3}{2}x = 7\]
Here again, the highest power of variable is one, Therefore option (c) is also incorrect.
Now let us check for option (d):
The equation given is-
\[{x^3}{x^2} = {\left( x \right)^3}\]
Or \[{x^2} = \dfrac{{{x^3}}}{{{x^3}}} = 1\]
$\Rightarrow$\[{x^2} = 1\]
Or \[{x^2} - 1 = 0\]
We can see that in this equation, the highest power of the variable is two, Therefore option (d) is our required quadratic equation.

Hence, the correct option is D

Note: When you change the side of any term, the sign must also change. If in any equation, the coefficient of $x^2$ is zero, that term is considered as zero and equation is no longer quadratic. Always follow the BODMAS rule i.e. bracket must be simplified first.