Question
Factor completely
5x^2-51x+10
5x^2-51x+10
Answers
Answered by
GPT-4o mini
To factor the quadratic expression \( 5x^2 - 51x + 10 \), we can use the method of finding two numbers that multiply to the product of the coefficient of \( x^2 \) (which is 5) and the constant term (which is 10), and sum to the coefficient of \( x \) (which is -51).
1. First, calculate the product of the coefficients:
\[
5 \times 10 = 50.
\]
2. We need to find two numbers that multiply to \( 50 \) and add to \( -51 \). The pair that meets these conditions is \( -50 \) and \( -1 \), since:
\[
-50 \times -1 = 50 \quad \text{and} \quad -50 + (-1) = -51.
\]
3. Next, we can rewrite the quadratic expression using these numbers:
\[
5x^2 - 50x - 1x + 10.
\]
4. Now group the terms:
\[
(5x^2 - 50x) + (-1x + 10).
\]
5. Factor out the common factors from each group:
\[
5x(x - 10) - 1(x - 10).
\]
6. Now, we factor out the common binomial factor \( (x - 10) \):
\[
(5x - 1)(x - 10).
\]
Thus, the completely factored form of the expression \( 5x^2 - 51x + 10 \) is:
\[
\boxed{(5x - 1)(x - 10)}.
\]
1. First, calculate the product of the coefficients:
\[
5 \times 10 = 50.
\]
2. We need to find two numbers that multiply to \( 50 \) and add to \( -51 \). The pair that meets these conditions is \( -50 \) and \( -1 \), since:
\[
-50 \times -1 = 50 \quad \text{and} \quad -50 + (-1) = -51.
\]
3. Next, we can rewrite the quadratic expression using these numbers:
\[
5x^2 - 50x - 1x + 10.
\]
4. Now group the terms:
\[
(5x^2 - 50x) + (-1x + 10).
\]
5. Factor out the common factors from each group:
\[
5x(x - 10) - 1(x - 10).
\]
6. Now, we factor out the common binomial factor \( (x - 10) \):
\[
(5x - 1)(x - 10).
\]
Thus, the completely factored form of the expression \( 5x^2 - 51x + 10 \) is:
\[
\boxed{(5x - 1)(x - 10)}.
\]