# What is the property called where an operator does not change the real roots of an equation?

Consider the two equations below set equal to each other: $$-x^2-3 x-2=-\frac{x^2}{3}-x-\frac23$$

The roots for both are the same but their graphs are different. They can trivially be converted to each other by dividing by 3 or multiplying by 3 respectively. I'm assuming this property where the roots do not change under a given operation has a name. What gives an operation this property? I have heard the term linear operator, is that what this is?

Edit: my copy-paste had a typo of $x^{2/3}$ where what was meant was $\frac{x^2}3$

Edit2: I guess what I am trying to ask is, in solving algebra problems certain operations are permitted that change the graph of the equation, but do not change the answer set. I want to know what that property is called.

• No, the above does not look like a linear operator. And are you sure they have the same roots? I'm not so sure... – Simply Beautiful Art Dec 28 '16 at 0:56
• I think OP meant to say $\frac{-x^2}{3}$ – cool.coolcoolcool Dec 28 '16 at 0:59
• @cool.coolcoolcool I would imagine so, but the OP has no changed anything. – Simply Beautiful Art Dec 28 '16 at 1:02
• I've heard the equations $-x^2 - 3x - 2 = 0$ and $-x^2/3 - x - 2/3$ called "equivalent," meaning they have the same solution set. I'm not sure about the "operator" part of your question though. – pjs36 Dec 28 '16 at 1:08
• Just to list a few operators that have this property, $f(x),\sin(f(x)),\tan(f(x)),e^{f(x)}-1,\text{ and}\ln(1+f(x))$ all have the same roots. – AlgorithmsX Dec 28 '16 at 1:11

Note that you can factor the LHS as $-1 * (x + 2)(x + 1)$, and the RHS as $-\frac{1}{3} * (x + 2)(x + 1)$. The key thing to notice here is that both polynomials will have the same roots, as the terms of form $(x + a_n)$ are identical; the only difference is the constant out in front.
We can now generalize this. Take a quadratic equation, whose factorization yields the equation $a * (x + r)(x + s)$. Note that if we divide or multiply by a constant, we will just change the value of $a$, and the roots will stay the same.
To address your second question, a linear operator is (and this is a very informal definition) an operator which preserves two important conditions: 1) when applied to the sum of two objects, it acts the same as if it was applied to both objects separately then they were added together, and 2) when applied to the product of an object and a number, it acts the same as if it were applied to the object and then multiplied by the number. Mathematically, this is expressed by $f(x + y) = f(x) + f(y), f(ax) = a * f(x)$.
As we're considering the multiplication/division operations here, we see that they are linear operators. If we multiply the sum of two numbers by a third number, it is equivalent to multiplying individually then summing. Also, if we multiply a product of two numbers by a third number, we can switch the order of multiplication and still receive the same answer. In other words, $a * (x + y) = ax + ay, a*(bx) = (ax)*b$.