The lengths of the sides of three squares are s, s+1, and s+2 If their total area is 365 cm^2, f

The lengths of the sides of three squares are s, s+1, and...

Key Concepts

Area of a Square

The area of a square is a fundamental concept in geometry and is determined by squaring its side length. In the context of the problem, calculating the areas of three squares with sides expressed in terms of a variable and summing these areas illustrates how geometric formulas are applied in algebraic expressions.

Perimeter of a Square

Understanding that the perimeter of a square is found by multiplying its side length by four is key. After determining the side lengths from the area equation, this concept is used to calculate the total perimeter by summing the perimeters of each individual square.

Square Geometry

This concept involves understanding that a square is a regular quadrilateral with four equal sides and right angles. In problems like this, the properties of squares—specifically that its area is calculated as the square of its side and its perimeter as four times the side—are crucial for setting up and solving equations related to the square's dimensions.

Quadratic Equations

When the areas of the squares are expressed in terms of a variable and summed to equal a given total, the resulting equation often becomes quadratic. Solving this quadratic equation to find the variable is an essential algebraic skill and highlights the interplay between algebra and geometry in problem-solving.

Transcript

00:01 We're given the sum of the areas of three squares is 365 square centimeters.

00:11 So total area, at, 365 square centimeters.

00:22 Now, we are also given the side length for the three squares.

00:26 First one has a side length s.

00:29 The second one has a side length s plus one.

00:33 And the third square has a side length s plus 2.

00:39 So now we can derive the formulas for the area of the three different triangles.

00:45 First one has a side of s, so its formula we'll call a1.

00:51 Let's simply be s squared.

00:55 The second triangle, a2, we'll have formula s plus 1 squared.

01:05 So we use a distributive property here.

01:07 Do s plus 1 times s plus 1 we get s squared plus 2 s plus 1 and then we get the third triangle third square a 3 s plus 2 squared similarly we distribute s plus 2 times s plus 2 and here we get s squared plus 4s plus 4.

01:58 So we now know that these three added together give us their total area.

02:03 So a1 plus a2 plus a3 is equal to the total area.

02:19 So that gives us s squared plus s squared plus 2s plus 2s plus 2s plus.

02:37 Plus 1 plus s squared plus 4 s plus 4 s plus 4 is equal to 3 65 square centimeters.

02:55 So let's combine our like terms.

02:59 We have s squared, we have 2 s 4 s and then we have our constant terms 1 and 4.

03:09 So combining our like terms, we now get 3s squared plus 6s plus 5 equal to 365.

03:31 So what we do next here, we take away 365 from both sides.

03:38 You have to use a little bit of factoring...