**Topic**

Given the equation \(a{x^2} + bx + c = 0\).

a) Consider the proposition “If \(a + b + c = 0\) then the equation \(a{x^2} + bx + c = 0\) has a solution equal to 1”. Is this statement true or false?

b) State the converse of the above proposition. Is the reverse statement true or false?

c) State the necessary input conditions for the equation \(a{x^2} + bx + c = 0\) to have a solution equal to 1.

**Solution method – See details**

The inverse of \(P \Rightarrow Q\) is \(Q \Rightarrow P\).

If \(P \Rightarrow Q\) and \(Q \Rightarrow P\) are both true, then we have the equivalent statement \(P \Leftrightarrow Q\), which can be stated in the form: “The input condition is sufficient to have P is Q”

**Detailed explanation**

a) This statement is true.

\(a + b + c = 0\) or \(a{.1^2} + b.1 + c = 0\), so \(x = 1\) is the solution of the equation \(a{ x^2} + bx + c = 0\).

b) Inverse proposition: “If the equation \(a{x^2} + bx + c = 0\) has a solution equal to 1, then \(a + b + c = 0\)”.

This inverse proposition is correct.

c) The input condition is sufficient for the equation \(a{x^2} + bx + c = 0\) to have a solution equal to 1, which is \(a + b + c = 0\).