Factor completely: 3f² + 8f - 3.
Understand the Problem
The question is asking to factor the quadratic expression completely, specifically the expression 3f² + 8f - 3.
Answer
The factored form is $(3f - 1)(f + 3)$.
Answer for screen readers
The completely factored form of the expression is $(3f - 1)(f + 3)$.
Steps to Solve
- Identify the coefficients
For the quadratic expression $3f^2 + 8f - 3$, identify the coefficients:
- $a = 3$ (coefficient of $f^2$)
- $b = 8$ (coefficient of $f$)
- $c = -3$ (constant term)
- Calculate the product $ac$
Multiply the coefficient of $f^2$ by the constant term:
$$ ac = 3 \times (-3) = -9 $$
- Find two numbers that multiply to $ac$ and add to $b$
Look for two numbers that multiply to $-9$ (the value of $ac$) and add to $8$ (the value of $b$). The numbers are $9$ and $-1$ because:
- $9 \times (-1) = -9$
- $9 + (-1) = 8$
- Rewrite the middle term using the two numbers
Rewrite the expression by splitting the middle term $8f$ into $9f - f$:
$$ 3f^2 + 9f - f - 3 $$
- Factor by grouping
Group the terms:
$$ (3f^2 + 9f) + (-f - 3) $$
Factor out the common terms in each group:
$$ 3f(f + 3) - 1(f + 3) $$
- Combine the factors
Now, factor out the common factor $(f + 3)$:
$$ (3f - 1)(f + 3) $$
The completely factored form of the expression is $(3f - 1)(f + 3)$.
More Information
Factoring quadratics is useful for solving equations, graphing parabolas, and finding roots. The product-sum method can help quickly find pairs of factors that satisfy the required conditions.
Tips
- Forgetting to multiply the leading coefficient $a$ with the constant $c$ when searching for the product.
- Incorrectly identifying the two numbers that both multiply to $ac$ and add to $b$. It's important to double-check both conditions.
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