Question

Simplify \[{(x + 2)^3} = 2x({x^2} - 1)\;\] and check whether its is a quadratic equation.

Answer 1

Hint: To check the given equation is quadratic equation or not just simplify the given equation first then try to compare the standard form of quadratic equation. If it exactly matches the standard format of the quadratic equation then the given equation will be a quadratic equation.

Complete step-by-step answer:
The given equation is \[{(x + 2)^3} = 2x({x^2} - 1)\;\] we have to check if the given equation is a quadratic equation or not?
We know that the general form of the quadratic equation is \[a{x^2} + bx + c = 0\] .
 The given equation is \[{(x + 2)^3} = 2x({x^2} - 1)\;\] can be simplified as follows:
By using the \[{(a + b)^3}\] formula (∵( \[{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\]
We get,
 \[{\left( {x + 2} \right)^3} = 2x\left( {{x^2} - 1} \right) \Rightarrow {x^3} + {2^3} + \left( {3 \times x \times 2} \right)\left( {x + 2} \right) = 2{x^3} - 2x\]
On simplifying after applying the formula we get,
 \[ \Rightarrow {x^3} + 8 + 6x\left( {x + 2} \right) = 2{x^3} - 2x\]
Or,
 \[ \Rightarrow {x^3} + 8 + 6{x^2} + 12x = 2{x^3} - 2x\]
Or,
 \[ \Rightarrow {x^3} + 8 + 6{x^2} + 12x - 2{x^3} + 2x = 0\]
or
 \[ \Rightarrow - {x^3} + 6{x^2} + 14x + 8 = 0\]
Since the variable x in the equation \[ - {x^3} + 6{x^2} + 14x + 8 = 0\] has degree 3, and we know that the quadratic equation should have highest degree 2 with coefficient of highest degree term not equal to zero. therefore, it is not of the form \[a{x^2} + bx + c = 0\] .
Hence, the equation \[{(x + 2)^3} = 2x({x^2} - 1)\;\] is not a quadratic equation.
So, the correct answer is “ the equation \[{(x + 2)^3} = 2x({x^2} - 1)\;\] is not a quadratic equation.”.

Note: The result we get after simplification of the given equation we get a polynomial that is cubic polynomial because it has the highest degree is equal to 3. Hence it is not a quadratic equation.