Animated Solution for Mathematics - Quadratic Equations: If $(1-p)$ is a root of quadratic equation $x^2+px+(1-p)=0$ then its root are

We are tasked with solving the quadratic equation $x^2+px+(1-p)=0$, given that $(1-p)$ is one of its roots. Our goal is to determine the specific values of these roots.

A root of an equation is a value that, when substituted for $x$, satisfies the equation. By substituting $x=1-p$ into the given quadratic equation, we establish the following identity:

Substituting $x=1-p$ yields:

While one could expand the terms, it is far more efficient to recognize the common factor of $(1-p)$ present in every term. By factoring out $(1-p)$, we simplify the expression significantly:

Inside the square brackets, the terms $-p$ and $+p$ cancel each other out perfectly. This leaves us with the constant sum $1+1=2$.

The equation simplifies to:

Since $2 eq 0$, we conclude that $1-p=0$, which implies that the parameter $p=1$.

With the value of $p$ determined, we substitute $p=1$ back into the original quadratic equation:

This results in the simplified quadratic equation:

Factoring the expression, we obtain:

Setting each factor to zero, we find the roots to be $x=0$ and $x=-1$.