Question

How do you solve the equation ${x^2} - 10x = 0$?

Answer 1

Hint: The given equation is a quadratic equation having constant term zero. Factorize it by taking the common term outside. Then equate those factors to zero to determine the root and solve the equation.

Complete step-by-step solution:
According to the question, we have an equation ${x^2} - 10x = 0$ and we have to show how to solve this equation.
As we can observe that this is a quadratic equation having constant term zero, we can solve this by factorization.
Since the equation is having only two terms on the left hand side, we’ll take the common factor of both the terms outside. Doing so, we’ll get:
 $ \Rightarrow x\left( {x - 10} \right) = 0$
So we have two factors of the equation. For finding the roots, we’ll put both these factors equal to zero separately. By doing this we’ll get:
$ \Rightarrow x = 0{\text{ or }}\left( {x - 10} \right) = 0$
Solving it further, we’ll get:
$ \Rightarrow x = 0{\text{ or }}x = 10$

Thus the equation has two real roots i.e. $x = 0$ and $x = 10$

Additional Information: If the quadratic equation consists of three terms and it is in the form of $y = a{x^2} + bx + c$ then for its factorization we will have split the middle term (i.e. $ + bx$) into two different terms such that the magnitude of product of their coefficients must be equal to the magnitude of the product of first and last coefficients (i.e. $a$ and $c$) in the quadratic equation.

Note: If we are facing any difficulty solving a quadratic equation using factorization method, we can also use a direct formula to find its roots. Let the quadratic equation be:
$ \Rightarrow y = a{x^2} + bx + c$
The formula to determine its roots is:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$