Question
Solve 5n^2-15n+10=0 by factoring.
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n=
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Answers
Answered by
GPT-4o mini
To solve the quadratic equation \( 5n^2 - 15n + 10 = 0 \) by factoring, we first factor out the greatest common factor (GCF), which is 5:
\[
5(n^2 - 3n + 2) = 0
\]
Next, we focus on factoring the quadratic expression \( n^2 - 3n + 2 \). We need to find two numbers that multiply to \( 2 \) (the constant term) and add to \( -3 \) (the coefficient of the linear term). The numbers that satisfy these conditions are \( -1 \) and \( -2 \).
Thus, we can factor \( n^2 - 3n + 2 \) as:
\[
(n - 1)(n - 2) = 0
\]
Now, substituting back into the equation gives us:
\[
5(n - 1)(n - 2) = 0
\]
Setting each factor equal to zero, we find:
1. \( n - 1 = 0 \) → \( n = 1 \)
2. \( n - 2 = 0 \) → \( n = 2 \)
Therefore, the solutions to the equation \( 5n^2 - 15n + 10 = 0 \) are:
\[
n = 1, 2
\]
Final answer:
\( n = 1, 2 \)
\[
5(n^2 - 3n + 2) = 0
\]
Next, we focus on factoring the quadratic expression \( n^2 - 3n + 2 \). We need to find two numbers that multiply to \( 2 \) (the constant term) and add to \( -3 \) (the coefficient of the linear term). The numbers that satisfy these conditions are \( -1 \) and \( -2 \).
Thus, we can factor \( n^2 - 3n + 2 \) as:
\[
(n - 1)(n - 2) = 0
\]
Now, substituting back into the equation gives us:
\[
5(n - 1)(n - 2) = 0
\]
Setting each factor equal to zero, we find:
1. \( n - 1 = 0 \) → \( n = 1 \)
2. \( n - 2 = 0 \) → \( n = 2 \)
Therefore, the solutions to the equation \( 5n^2 - 15n + 10 = 0 \) are:
\[
n = 1, 2
\]
Final answer:
\( n = 1, 2 \)