Question

Solve \(3(1-x)^{2}-2(1-x)-1=0\)

Answer 1

Let \(1-x =y\)

so the equation \(3(1-x)^{2}-2(1-x)-1=0\) becomes:

\[3y^{2}-2y-1=0\]

rewrite the left hand side of th equation:

\[3y^{2}-3y+y-1=0\]

Now factorize with the common factors

\[3y(y-1)+1(y-1)=0\]

Since \(y-1\) is a common factor we have:

\[(3y+1)(y-1)=0\]

which implies that

\(3y+1=0\) or \(y-1=0\)

solving the two linear equations gives:

\(y=\frac{-1}{3}\) or \(y=1\)

But remember that \(y=1-x\), so:

\(1-x=\dfrac{-1}{3}\) or \(1-x=1\)

\[\therefore x=0 \quad \text{OR} \quad x=\dfrac{4}{3}\]