https://file.aiquickdraw.com/mathbot/file/1730376775132o7juff9q.jpg

The expression in the image is:

\[
5 \left( t^2 + \frac{8}{5}t + \left(\frac{4}{5}\right)^2 - \left(\frac{4}{5}\right)^2 + 1 \right)
\]

To simplify or prove this is equivalent to \( 5t^2 + 8t + 5 \), let's go through the steps:

1. **Rewrite the expression in the brackets:**
   \[
   t^2 + \frac{8}{5}t + \left(\frac{4}{5}\right)^2 - \left(\frac{4}{5}\right)^2 + 1
   \]
   Notice that \( \left(\frac{4}{5}\right)^2 \) was added and subtracted inside the parentheses, likely to complete the square.

2. **Complete the square:**
   The terms \( t^2 + \frac{8}{5}t + \left(\frac{4}{5}\right)^2 \) form a perfect square trinomial:
   \[
   \left(t + \frac{4}{5}\right)^2
   \]
   So the expression inside the parentheses becomes:
   \[
   \left(t + \frac{4}{5}\right)^2 - \left(\frac{4}{5}\right)^2 + 1
   \]

3. **Simplify further:**
   Substitute \( \left(t + \frac{4}{5}\right)^2 - \left(\frac{4}{5}\right)^2 \) and simplify with the \( +1 \) term:
   \[
   = \left(t + \frac{4}{5}\right)^2 - \frac{16}{25} + 1
   \]
   Convert 1 to a fraction with a denominator of 25:
   \[
   = \left(t + \frac{4}{5}\right)^2 - \frac{16}{25} + \frac{25}{25}
   \]
   \[
   = \left(t + \frac{4}{5}\right)^2 + \frac{9}{25}
   \]

4. **Multiply by 5:**
   Now, distributing the 5 outside the parentheses:
   \[
   5 \left(\left(t + \frac{4}{5}\right)^2 + \frac{9}{25}\right)
   \]
   \[
   = 5 \left(t + \frac{4}{5}\right)^2 + 5 \cdot \frac{9}{25}
   \]
   \[
   = 5 \left(t + \frac{4}{5}\right)^2 + \frac{45}{25}
   \]
   Simplify \(\frac{45}{25}\):
   \[
   = 5t^2 + 8t + 5
   \]

Thus, we have shown that the expression simplifies to \( 5

5 + 5t^2 + 8t. Prove the above equation in the photo.

Explore how to prove and simplify the quadratic expression 5 + 5t^2 + 8t by completing the square. This step-by-step solution demonstrates algebraic techniques to simplify the equation, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation