Factor $$ b^{2}-2 b^{2}+3 b-6 $$

Question

UpStudy AI · Accepted Answer

Let's factor the expression $$ b^{2} - 2b^{2} + 3b - 6 $$.

### Step 1: Simplify the Expression
Combine like terms:
$$
b^{2} - 2b^{2} + 3b - 6 = -b^{2} + 3b - 6
$$

### Step 2: Factor Out the Common Factor
Notice that each term has a common factor of $$-1$$:
$$
- b^{2} + 3b - 6 = - (b^{2} - 3b + 6)
$$

### Step 3: Check for Further Factorization
Now, consider the quadratic inside the parentheses: $$ b^{2} - 3b + 6 $$.

To factor this quadratic, we look for two numbers that multiply to $$6$$ (the constant term) and add up to $$-3$$ (the coefficient of $$b$$). However, there are no such integer pairs.

If you attempt to factor it over the real numbers, you'll find that the discriminant is negative:
$$
\Delta = (-3)^2 - 4 \times 1 \times 6 = 9 - 24 = -15
$$
Since the discriminant is negative, the quadratic does not factor over the real numbers.

### Final Factored Form
Thus, the expression cannot be factored further using real numbers. The fully factored form is:
$$
- (b^{2} - 3b + 6)
$$

**Answer:**  
After simplifying, the factored form is – (b² – 3 b + 6):
$$
- (b^{2} - 3b + 6)
$$
The expression $$ b^{2} - 2b^{2} + 3b - 6 $$ simplifies to $$ -b^{2} + 3b - 6 $$. This quadratic does not factor further over the real numbers.

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Solución

Step 1: Simplify the Expression

Step 2: Factor Out the Common Factor

Step 3: Check for Further Factorization

Final Factored Form

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