Factor completely: 3f² + 8f - 3

Understand the Problem

The question is asking to completely factor the quadratic expression 3f² + 8f - 3. This involves finding two binomials that multiply to this quadratic expression.

Answer

The completely factored form is $(3f - 1)(f + 3)$.

Steps to Solve

Identify the coefficients The quadratic expression is of the form $ax^2 + bx + c$, where:

$a = 3$
$b = 8$
$c = -3$

Multiply $a$ and $c$ Now, multiply the coefficients $a$ and $c$: $$ 3 \times (-3) = -9 $$
Find two numbers that multiply to $ac$ and add to $b$ Next, find two numbers that multiply to $-9$ (our product) and add to $8$ (our sum). The numbers are $9$ and $-1$: $$ 9 \times (-1) = -9 $$ $$ 9 + (-1) = 8 $$
Rewrite the middle term using these numbers Now we will rewrite the quadratic expression by replacing $8f$ with $9f - f$: $$ 3f^2 + 9f - f - 3 $$
Factor by grouping Group the terms: $$ (3f^2 + 9f) + (-f - 3) $$ Now factor out the common terms: $$ 3f(f + 3) - 1(f + 3) $$
Factor out the common binomial Now, factor out the common binomial $(f + 3)$: $$ (3f - 1)(f + 3) $$

The completely factored form of the quadratic expression is $(3f - 1)(f + 3)$.

More Information

Factoring quadratics like this is a critical skill in algebra. It helps in simplifying expressions and solving equations. The process involves breaking down a quadratic into its components, making it easier to work with.

Tips

Forgetting to check that the two numbers truly multiply and add correctly.
Not properly factoring by grouping, leading to incorrect binomials.

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