Question

Is the following equation quadratic?
$ {x^2} - 2x + 5 = {x^2} $

Answer 1

Hint: The given problem requires us to recognize if the equation provided to us in the problem is a quadratic equation or not. A quadratic equation can be defined as a polynomial equation of a second degree, which implies that it comprises a minimum of one term that is squared. Hence, the highest degree of variable in a quadratic equation is $ 2 $ . The general form of a quadratic equation in a variable x is $ a{x^2} + bx + c = 0 $ where a, b and c are numerical coefficients and the leading coefficient, that is a, is non-zero. We will first convert the equation provided to us into the general form of quadratic equation and then check for the conditions of a quadratic equation.

Complete step by step solution:
Now, the equation given to us is $ {x^2} - 2x + 5 = {x^2} $ .
We have to check whether the equation is quadratic or not. So, we convert the given equation into the general form of a quadratic equation.
The general form of a quadratic equation is $ a{x^2} + bx + c = 0 $ .
Now, shifting all the terms of the equation $ {x^2} - 2x + 5 = {x^2} $ to the left side of the equation in order to resemble the general form of a quadratic equation, we get,
 $ \Rightarrow {x^2} - 2x + 5 - {x^2} = 0 $
Cancelling the like terms with opposite signs, we get,
 $ \Rightarrow - 2x + 5 = 0 $
Now, we have only two terms in the equation obtained. Also, the highest degree of the variable x in the equation obtained is $ 1 $ . So, the coefficient of the squared term would be zero even if we try to introduce the squared term in the equation. So, we get,
 $ \Rightarrow 0{x^2} - 2x + 5 = 0 $
Now, for an equation to be quadratic, it should be of the form $ a{x^2} + bx + c = 0 $ where a, b and c are numerical coefficients and the leading coefficient, that is a, is non-zero. The highest degree of the variable in a quadratic variable is $ 2 $ .
Now, we have the leading coefficient of the equation $ {x^2} - 2x + 5 = {x^2} $ when represented in the form $ a{x^2} + bx + c = 0 $ as zero. Also, the highest degree of the variable is $ 1 $ when the given equation is represented in the general form of the equation.
So, the equation $ {x^2} - 2x + 5 = {x^2} $ is not a quadratic equation.

Note: Quadratic equations are the polynomial equations with degree of the variable or unknown as $ 2 $ . The coefficient of the term consisting of $ {x^2} $ should not be zero. One must know the standard form of quadratic equations and how to convert a given equation into the standard form in order to solve such questions. We must know the transposition rule so as to simplify the equation by shifting the terms from one side of the equation to the other. We also know that we can solve the quadratic equation by factorization method or by quadratic formula.