Asked by pp
Factor Completely
2, b, squared, minus, 18, b, plus, 40
2b
2
−18b+40
2, b, squared, minus, 18, b, plus, 40
2b
2
−18b+40
Answers
Answered by
GPT-4o mini
To factor the expression \( 2b^2 - 18b + 40 \) completely, we can follow these steps:
1. **Factor out the greatest common factor (GCF)**:
First, we notice that there is a common factor of 2 in each term:
\[
2b^2 - 18b + 40 = 2(b^2 - 9b + 20)
\]
2. **Factor the quadratic expression**:
Next, we will factor the quadratic \( b^2 - 9b + 20 \). We need to find two numbers that multiply to 20 (the constant term) and add up to -9 (the coefficient of \( b \)).
The numbers that work are -4 and -5, since:
\[
-4 \times -5 = 20 \quad \text{and} \quad -4 + (-5) = -9
\]
So we can factor the quadratic:
\[
b^2 - 9b + 20 = (b - 4)(b - 5)
\]
3. **Combine with the GCF**:
Now, substitute this factorization back into our expression:
\[
2(b^2 - 9b + 20) = 2(b - 4)(b - 5)
\]
Thus, the completely factored form of the expression \( 2b^2 - 18b + 40 \) is:
\[
\boxed{2(b - 4)(b - 5)}
\]
1. **Factor out the greatest common factor (GCF)**:
First, we notice that there is a common factor of 2 in each term:
\[
2b^2 - 18b + 40 = 2(b^2 - 9b + 20)
\]
2. **Factor the quadratic expression**:
Next, we will factor the quadratic \( b^2 - 9b + 20 \). We need to find two numbers that multiply to 20 (the constant term) and add up to -9 (the coefficient of \( b \)).
The numbers that work are -4 and -5, since:
\[
-4 \times -5 = 20 \quad \text{and} \quad -4 + (-5) = -9
\]
So we can factor the quadratic:
\[
b^2 - 9b + 20 = (b - 4)(b - 5)
\]
3. **Combine with the GCF**:
Now, substitute this factorization back into our expression:
\[
2(b^2 - 9b + 20) = 2(b - 4)(b - 5)
\]
Thus, the completely factored form of the expression \( 2b^2 - 18b + 40 \) is:
\[
\boxed{2(b - 4)(b - 5)}
\]