Factor h=-16t^2+16t+32
The question is asking for the process of factoring the quadratic equation h = -16t^2 + 16t + 32. Factoring an equation like this means breaking it down into simpler expressions (factors) that multiply together to give the original equation. The goal is to express h as a product of two or more factors, typically involving the variable t in this case, which when combined using multiplication will recreate the original quadratic equation.
$h = - 16 t^{2} + 16 t + 32$
Answer
Solution:
Step 1: Extract the common factor
Step 1.1: Extract the common factor from the first term
Extract $16$ from $-16t^2$. This gives us $h = 16(-t^2) + 16t + 32$.
Step 1.2: Extract the common factor from the second term
Extract $16$ from $16t$. This results in $h = 16(-t^2) + 16(t) + 32$.
Step 1.3: Extract the common factor from the third term
Extract $16$ from $32$. We have $h = 16(-t^2) + 16t + 16 \cdot 2$.
Step 1.4: Combine the extracted factors
Combine the terms with the common factor $16$. This gives $h = 16(-t^2 + t) + 16 \cdot 2$.
Step 1.5: Final extraction of the common factor
Extract $16$ from the entire expression. We obtain $h = 16(-t^2 + t + 2)$.
Step 2: Factor the quadratic expression
Step 2.1: Factor by grouping
Step 2.1.1: Rewrite the middle term
For a quadratic expression $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$. Here, $ac = -1 \cdot 2 = -2$ and $b = 1$.
Step 2.1.1.1: Multiply by $1$
We have $h = 16(-t^2 + 1t + 2)$.
Step 2.1.1.2: Split the middle term
Split $1$ into $-1 + 2$. This gives $h = 16(-t^2 + (-1 + 2)t + 2)$.
Step 2.1.1.3: Apply distributive property
Distribute the terms to get $h = 16(-t^2 - 1t + 2t + 2)$.
Step 2.1.2: Factor out the greatest common factor from each group
Step 2.1.2.1: Group terms
Group the first two and last two terms together. This results in $h = 16((-t^2 - 1t) + (2t + 2))$.
Step 2.1.2.2: Factor out the GCF from each group
Factor out the greatest common factor from each group to get $h = 16(t(-t - 1) + 2(-t - 1))$.
Step 2.1.3: Factor the polynomial
Factor out the common binomial factor $-t - 1$. This gives $h = 16(-t - 1)(t - 2)$.
Step 2.2: Simplify the expression
Remove unnecessary parentheses to get the final factored form $h = 16(-t - 1)(t - 2)$.
Knowledge Notes:
To factor a quadratic expression of the form $ax^2 + bx + c$, one can employ several methods, including:
Extracting common factors: If each term in the quadratic expression has a common factor, it can be factored out to simplify the expression.
Factoring by grouping: This method involves splitting the middle term into two terms that can be grouped in such a way that each group has a common factor.
Factoring quadratics: For a quadratic expression $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add up to $b$. These two numbers are used to split the middle term and facilitate factoring by grouping.
Distributive property: Also known as the factorization property, it allows us to factor out a common factor from a sum or difference of terms.
In the given problem, we first extracted the common factor of $16$ from each term. Then, we applied the method of factoring by grouping to the quadratic expression inside the parentheses. After splitting the middle term and grouping, we factored out the greatest common factor from each group, which led us to the final factored form of the expression.
