Question
Solve the equation 0=3x2−4x+1
by factoring. One of the solutions is provided. What is the other solution?(1 point)
The solutions are x=13
and x=
.
by factoring. One of the solutions is provided. What is the other solution?(1 point)
The solutions are x=13
and x=
.
Answers
GPT-4o mini
To solve the equation \( 0 = 3x^2 - 4x + 1 \) by factoring, we first need to find two numbers that multiply to the product of the coefficient of \( x^2 \) (which is 3) and the constant term (which is 1), giving us \( 3 \cdot 1 = 3 \), and that also add to the coefficient of \( x \) (which is -4).
The two numbers that multiply to 3 and add to -4 are -3 and -1.
We can rewrite the middle term of the quadratic equation using these numbers:
\[
3x^2 - 3x - x + 1 = 0
\]
Now, we can factor by grouping:
\[
3x(x - 1) - 1(x - 1) = 0
\]
This can be factored as:
\[
(3x - 1)(x - 1) = 0
\]
Now, we set each factor equal to zero:
1. \( 3x - 1 = 0 \)
\( 3x = 1 \)
\( x = \frac{1}{3} \)
2. \( x - 1 = 0 \)
\( x = 1 \)
Thus, the solutions to the equation are \( x = \frac{1}{3} \) and \( x = 1 \).
Since one of the solutions provided is \( x = 1 \), the other solution is:
\[
x = \frac{1}{3}
\]
The two numbers that multiply to 3 and add to -4 are -3 and -1.
We can rewrite the middle term of the quadratic equation using these numbers:
\[
3x^2 - 3x - x + 1 = 0
\]
Now, we can factor by grouping:
\[
3x(x - 1) - 1(x - 1) = 0
\]
This can be factored as:
\[
(3x - 1)(x - 1) = 0
\]
Now, we set each factor equal to zero:
1. \( 3x - 1 = 0 \)
\( 3x = 1 \)
\( x = \frac{1}{3} \)
2. \( x - 1 = 0 \)
\( x = 1 \)
Thus, the solutions to the equation are \( x = \frac{1}{3} \) and \( x = 1 \).
Since one of the solutions provided is \( x = 1 \), the other solution is:
\[
x = \frac{1}{3}
\]