Question

How do you solve ${x^3} - 5{x^2} = 0$ ?

Answer 1

Hint: To solve this we have to know about algebraic identity.
Cubic equation: a polynomial equation in which the highest sum of exponents of a variable in any term is three. The value of $x$ for which the result of the equation is zero then that value will be the root of the cubic equation.

Complete step-by-step solution:
To solve this question. We will first factorize it.
We try to take common from the given equation ${x^3} - 5{x^2} = 0$ .
Take ${x^2}$ as common from both terms. We get,
 $ \Rightarrow {x^2}\left( {x - 5} \right) = 0$
Hence LHS is equal to zero so, we can conclude that either,
 $ \Rightarrow {x^2} = 0$
Or $ \Rightarrow \left( {x - 5} \right) = 0$
So, we will get,
 $ \Rightarrow {x^2} = 0$
 $ \Rightarrow x = 0$
And $\left( {x - 5} \right) = 0$
 $ \Rightarrow x = 5$
The given equation is the cubic equation.

So, the roots of equation ${x^3} - 5{x^2} = 0$ is $0\,and\,5$ .

Note: Application of cubic equation is angle trisection and doubling the cube are two ancient problems of geometry that have been proved to not be solvable by straightedge and compass construction, because they are equivalent to solving a cubic equation.