Let's verify if your approach and equation are correct.

Problem Summary

The rectangular picture frame has:

• Width $w$
• Length $w + 3$
• Area of glass in the frame: $154 \, \text{cm}^2$

Solution Verification

1. Formulating the Area Equation: The area $A$ of a rectangle is given by: $A = \text{length} \times \text{width}$ Substituting the given dimensions: $154 = w \times (w + 3)$

2. Expanding the Equation: Expanding $w \times (w + 3)$: $154 = w^2 + 3w$

3. Setting the Equation to Standard Form: To set it to standard form, move 154 to the other side: $0 = w^2 + 3w - 154$

Conclusion

Your equation, $0 = w^2 + 3w - 154$, is correct and matches the standard form of a quadratic equation.

Would you like further details on solving this quadratic equation?

Here are 5 related questions for deeper understanding:

1. How would you solve the equation $w^2 + 3w - 154 = 0$ for $w$?
2. What are the possible values of $w$ if you solve the quadratic equation?
3. How does changing the dimensions $w$ and $w + 3$ affect the area of the frame?
4. How could you verify the solutions to ensure they satisfy the original area equation?
5. What other methods could you use to solve a quadratic equation like this?

Tip: To solve a quadratic equation, consider using the quadratic formula: $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 3$, and $c = -154$.