Question

Check whether the following is a quadratic equation or not ${(x + 2)^3} = {x^3} - 4$.

Answer 1

First of all, we should know about the quadratic equation. The general form of the quadratic equation is $a{x^2} + bx + c = 0$ where a,b, and c are the Constants. The Highest Power of $x$ in this equation is $2$. In this equation, the coefficient of $x$ and constant may or may not be zero. The Coefficient of ${x^2}$ must not be zero. If the Coefficient of ${x^2}$ is equal to zero, that is not a Quadratic equation. The key point in this problem is that the coefficient of ${x^2}$ should not be zero and the highest power of $x$ is $2$.

Complete step by step solution:
We know that in a Quadratic equation, the Highest Power of $x$ must be $2$. Now take the problem.
${(x + 2)^3} = {x^3} - 4$
We have to check whether it is a quadratic equation or not. Let us simplify the Problem by Using Algebraic Formula and find out the final Equation.
If the highest Power of $x$ in that final equation is Zero, Then it is a Quadratic equation.
${(x + 2)^3} = {x^3} - 4$
Let Us Simplify the above Equation by using the Algebraic Formula
${(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} \\\ {(x + 2)^3} = {x^3} - 4 \\\ {x^3} + 6{x^2} + 12x + 8 = {x^3} - 4 \\\$
Simplify Again
${x^3} - {x^3} + 6{x^2} + 12x + 8 + 4 = 0 \\\ 0{x^3} + 6{x^2} + 12x + 12 = 0 \\\ 6{x^2} + 12x + 12 = 0 \\\$
In the above Equation, The Highest Power of $x$ in this equation is $2$.
The coefficient of ${x^2}$ is not Zero. Hence, it is a Quadratic Equation.

Note:
If the highest power of the Equation is $3$, it is a Cubic Equation, not a Quadratic Equation. We have to check the Coefficient of the highest Power is Zero, neglect that term, and then check the Highest Power of $x$ in the Final equation. If the power is $2$, declare that equation as a quadratic equation. If the highest power of $x$ in the equation is $1$, that is the linear equation, not a quadratic equation.