25t^2+80t+64 in factores form k(at+b)(ct+d)

To factor \( 25t^2 + 80t + 64 \) into the form \( k(at + b)(ct + d) \), we first use the method of factoring trinomials.

We will proceed step by step:

### Step 1: Identify the coefficients
The given quadratic expression is:
\[
25t^2 + 80t + 64
\]
Here:
- The coefficient of \(t^2\) is \(25\),
- The coefficient of \(t\) is \(80\),
- The constant term is \(64\).

### Step 2: Check for common factors
There are no common factors in the three terms, so we proceed with factoring.

### Step 3: Apply the factoring method
We need to find factors of the first and last terms that multiply together to give the middle term (cross multiplication technique). We rewrite the middle term \(80t\) in a way that allows us to factor by grouping.

#### Step 3.1: Factor the trinomial
\[
25t^2 + 80t + 64 = (5t + 8)(5t + 8)
\]
So the factorized form is:
\[
(5t + 8)(5t + 8)
\]

### Step 4: Simplify and finalize
Since the factors are the same, the factorized form is:
\[
(5t + 8)^2
\]

Thus, the quadratic expression \( 25t^2 + 80t + 64 \) can be written in the form:
\[
(5t + 8)^2
\]

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### Related Questions:
1. How do you factor trinomials where the coefficient of \(t^2\) is not 1?
2. What is the method for factoring perfect square trinomials?
3. How do you apply the cross multiplication technique in factoring?
4. What are the conditions for a trinomial to be a perfect square?
5. How does factoring help solve quadratic equations?

**Tip:** Always check if a trinomial is a perfect square by taking the square root of the first and last terms!

25t^2 + 80t + 64 in factored form k(at+b)(ct+d)

Learn how to factor the quadratic trinomial 25t^2 + 80t + 64 into the form (5t + 8)^2. This detailed guide covers factoring techniques and the perfect square trinomial method, suitable for students in Grades 9-11.

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