Formulating a Quadratic Equation to Represent the Model

8.1.3 Formulating a Quadratic Equation to Represent the Model

8.1.3 • Formulating a Quadratic Equation to Represent the Model

Activity

Here is a diagram that shows the picture with a frame that is the same width all the way around.

The picture is 7 inches by 4 inches. The frame is created from 10 square inches of framing material (in the form of a rectangle measuring 4 inches by 2.5 inches).

A colorful illustration of various flowers with vibrant petals and green stems, set against a dark background with a light peach border.

Write an expression to represent the length of the framed picture on its shorter side. Use $x$ to represent the width of the frame.

Compare your answer:

$2 x + 4$

Write an expression to represent the length of the framed picture on its longer side. Use $x$ to represent the width of the frame.

Compare your answer:

$2 x + 7$

Use the expressions from the previous exercises to write an equation to represent the relationship between the measurements of the framed picture and its total area. Be prepared to explain what each part of your equation represents.

Compare your answer:

$(2 x + 7) (2 x + 4) = 38$ or $4 x^{2} + 22 x + 28 = 38$ (or equivalent)

A picture is shown with a frame of uniform width around it. The dimensions of the picture are 7 inches by 4 inches. The width of the frame is x. The total length of the framed picture is 2x + 7. The total width of the framed picture is 2x + 4.

What would a solution to this equation mean in this situation?

Compare your answer:

A solution would represent the width of the frame when all of the framing material is used.

Video: Formulating a Quadratic Equation to Represent the Model

Watch the following video to learn more about formulating a quadratic equation to represent the model.

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Self Check

Suppose a different picture is 5 inches by 5 inches. The frame is created from 11 square inches of framing material. The frame will be uniform in width, and all of the framing material is used. Which equation would you need to solve to determine the value of $x$ , the width of the frame?

A square picture has side lengths measuring 5 inches. It is surrounded by a square frame that is uniformly x inches in width.

$4x^2 + 20x = 36$
$x^2 + 25 = 11$
$x^2 + 10x + 25 = 36$
$4x^2 + 20x + 25 = 36$

Additional Resources

Formulating a Quadratic Equation to Represent the Model

Let’s look at a similar situation to the previous activities but using a picture with different measurements.

Imagine a picture 8 inches by 6 inches. The frame is created from 32 square inches of framing material. The frame will be uniform in width, and all of the framing material is used.

How do we write an equation representing the total area of the framed picture?

A simple illustration of brown mountains under an orange sky with a yellow sun and white clouds, framed by a yellow border.

Let’s break down the measurements that we know.

Let $x$ represent the thickness of the frame in inches.

A rectangular picture has side lengths measuring 8 inches by 6 inches. It is surrounded by a frame that is uniformly x inches in width.

Since the picture is 8 inches by 6 inches, it has an area of 48 square inches.

The frame is created from 32 square inches of framing material, and we must use all of it. So, the total area of the framed picture is $48 + 32$ , or 80 square inches.

The length of one side of the framed picture is $x + 6 + x = 2 x + 6$ .

A diagram representing a rectangular picture inside a rectangular picture frame is shownthe rectangular picture frame has a side length of 6 inches and x and x labeled.

The length of the other side of the framed picture is $x + 8 + x = 2 x + 8$ .

A diagram representing a rectangular picture inside a rectangular picture frame is shownthe rectangular picture frame has a side length of 8 inches and x and x labeled.

So, the area of the framed picture is $(2 x + 8) (2 x + 6)$ . We can set this equal to 80 square inches.

$(2 x + 8) (2 x + 6) = 80$

Let’s multiply.

$4 x^{2} + 12 x + 16 x + 48 = 80$

$4 x^{2} + 28 x + 48 = 80$

So, the equation representing the total area of the framed picture is $4 x^{2} + 28 x + 48 = 80$ .

This equation is a quadratic equation. A quadratic equation is defined as one that can be written in the form $a x^{2} + b x + c = 0$ , where $a$ is not 0.

Try it

Formulating a Quadratic Equation to Represent the Model

A picture measures 9 inches by 7 inches. Its frame is created from 36 square inches of framing material. The frame will be uniform in width, and all of the framing material is used.

Write an equation representing the total area of the framed picture.

Here is how to find the quadratic equation to represent the total area of the framed picture:

The area of the picture is 63 square inches. The total area of the framed picture will be $63 + 36$ , or 99 square inches.

The sides of the framed picture are represented by $(2 x + 9)$ and $(2 x + 7)$ .

So, the total area of the framed picture is represented by $4 x^{2} + 32 x + 63 = 99$ .