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

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Solution:

Step 1: Extract the common factor

Step 1.1: Extract the common factor from the first term

Extract $16$ from $- 16 t^{2}$ . This gives us $h = 16 (- t^{2}) + 16 t + 32$ .

Step 1.2: Extract the common factor from the second term

Extract $16$ from $16 t$ . 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 $h = 16 (- t^{2}) + 16 t + 16 \cdot 2$ .

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 $a x^{2} + b x + c$ , find two numbers that multiply to $a c$ and add to $b$ . Here, $a c = - 1 \cdot 2 = - 2$ and $b = 1$ .

Step 2.1.1.1: Multiply by $1$

We have $h = 16 (- t^{2} + 1 t + 2)$ .

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 $h = 16 (- t^{2} - 1 t + 2 t + 2)$ .

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 $h = 16 ((- t^{2} - 1 t) + (2 t + 2))$ .

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 $a x^{2} + b x + c$ , 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 $a x^{2} + b x + c$ , we look for two numbers that multiply to $a c$ 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.