Finding the Area of the Entire Deck

Oct 13, 2025 by talynt.org 79 views

Understanding the Original Deck Area

When building an addition to your deck, it's essential to consider the original deck area. In this case, the area of the original deck can be represented by the quadratic equation $y^2 + 2y - 15$ . This equation is a quadratic function in the form of $ax^2 + bx + c$ , where $a = 1$ , $b = 2$ , and $c = -15$ .

The Quadratic Formula

To find the value of $y$ , we can use the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . In this case, the values of $a$ , $b$ , and $c$ are already known. We can substitute these values into the quadratic formula to find the value of $y$ .

Solving for $y$

Substituting the values of $a$ , $b$ , and $c$ into the quadratic formula, we get:

$y = \frac{-2 \pm \sqrt{2^2 - 4(1)(-15)}}{2(1)}$

Simplifying the expression under the square root, we get:

$y = \frac{-2 \pm \sqrt{4 + 60}}{2}$

$y = \frac{-2 \pm \sqrt{64}}{2}$

$y = \frac{-2 \pm 8}{2}$

This gives us two possible values for $y$ :

$y = \frac{-2 + 8}{2} = 3$

$y = \frac{-2 - 8}{2} = -5$

Finding the Area of the Entire Deck

Now that we have found the value of $y$ , we can find the area of the entire deck after the addition. The area of the addition is given as 24 square yards.

The Area of the Entire Deck

To find the area of the entire deck, we need to add the area of the original deck to the area of the addition. The area of the original deck is represented by the quadratic equation $y^2 + 2y - 15$ . We can substitute the value of $y$ into this equation to find the area of the original deck.

Substituting the Value of $y$

Substituting the value of $y = 3$ into the equation $y^2 + 2y - 15$ , we get:

$3^2 + 2(3) - 15$

$= 9 + 6 - 15$

$= 0$

This means that the area of the original deck is 0 square yards.

The Area of the Entire Deck

Now that we have found the area of the original deck, we can add the area of the addition to find the area of the entire deck:

Area of the entire deck = Area of the original deck + Area of the addition

$= 0 + 24$

$= 24$

Therefore, the area of the entire deck after the addition is 24 square yards.

Conclusion

In this article, we have found the value of $y$ that represents the area of the original deck. We have also found the area of the entire deck after the addition by adding the area of the original deck to the area of the addition. The area of the entire deck is 24 square yards.

Final Answer

The final answer is $\boxed{24}$ .

Understanding the Original Deck Area

The Quadratic Formula

Solving for $y$

Substituting the values of $a$ , $b$ , and $c$ into the quadratic formula, we get:

$y = \frac{-2 \pm \sqrt{2^2 - 4(1)(-15)}}{2(1)}$

Simplifying the expression under the square root, we get:

$y = \frac{-2 \pm \sqrt{4 + 60}}{2}$

$y = \frac{-2 \pm \sqrt{64}}{2}$

$y = \frac{-2 \pm 8}{2}$

This gives us two possible values for $y$ :

$y = \frac{-2 + 8}{2} = 3$

$y = \frac{-2 - 8}{2} = -5$

Q&A

Q: What is the area of the original deck?

A: The area of the original deck is represented by the quadratic equation $y^2 + 2y - 15$ . We can substitute the value of $y$ into this equation to find the area of the original deck.

Q: How do I find the value of $y$ ?

A: To find the value of $y$ , we can use the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . In this case, the values of $a$ , $b$ , and $c$ are already known. We can substitute these values into the quadratic formula to find the value of $y$ .

Q: What is the area of the entire deck after the addition?

A: To find the area of the entire deck after the addition, we need to add the area of the original deck to the area of the addition. The area of the original deck is represented by the quadratic equation $y^2 + 2y - 15$ . We can substitute the value of $y$ into this equation to find the area of the original deck.

Q: How do I calculate the area of the entire deck?

A: To calculate the area of the entire deck, we need to add the area of the original deck to the area of the addition. The area of the original deck is represented by the quadratic equation $y^2 + 2y - 15$ . We can substitute the value of $y$ into this equation to find the area of the original deck.

Q: What is the final answer?

A: The final answer is $\boxed{24}$ .

Conclusion

Final Answer

The final answer is $\boxed{24}$ .

Additional Resources

Quadratic Formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Quadratic Equation: $y^2 + 2y - 15$
Area of the Entire Deck: $24$ square yards