Question

Solve the equation 0=3x2−4x+1 by factoring. One of the solutions is provided. What is the other solution?

Answer 1

To solve the equation \(0 = 3x^2 - 4x + 1\) by factoring, we can start by rewriting the equation into a factored form.

First, let's look for two numbers that multiply to \(ac\) (where \(a = 3\) and \(c = 1\), thus \(ac = 3 \cdot 1 = 3\)) and add up to \(b\) (where \(b = -4\)).

We need two numbers that multiply to \(3\) and add to \(-4\). The numbers that work are \(-3\) and \(-1\).

We can rewrite the middle term of the quadratic equation using these two numbers:

\[
3x^2 - 3x - 1x + 1 = 0
\]

Next, we can group the terms:

\[
(3x^2 - 3x) + (-1x + 1) = 0
\]

Now factor out the common factors in each group:

\[
3x(x - 1) - 1(x - 1) = 0
\]

This allows us to factor by grouping:

\[
(3x - 1)(x - 1) = 0
\]

Now we can set each factor equal to zero to find the solutions:

1. \(3x - 1 = 0\)
\[
3x = 1 \implies x = \frac{1}{3}
\]

2. \(x - 1 = 0\)
\[
x = 1
\]

Thus, the solutions to the equation \(0 = 3x^2 - 4x + 1\) are \(x = \frac{1}{3}\) and \(x = 1\).

Since one solution was given, we can conclude that the other solution is:

\[
\boxed{1}
\]