We will now use a quadratic equation to represent a contextual situation. There are many situations that can be modeled by a quadratic equation. Let's look at some.
Example 1: Given the rectangle whose area is \(32\) cm\(^2\) and the dimensions labeled below. Find the value of \(x\).
Example 1: Given the rectangle whose area is \(32\) cm\(^2\) and the dimensions labeled below. Find the value of \(x\).
Solution:
Start by setting up the equation using the area of a rectangle. To solve this equation, let's set one side needs equal to zero. Mutiply the two binomials (using the distributive property) Now subtract \(32\) from both sides of the equation. |
\((x + 2)(x - 8) = 32\) \(x^2 - 8x - 16 = 32\) \(x^2 - 6x - 16 - 32 = 32 - 32\) \(x^2 - 6x - 48 = 0\) |
Next we need to solve the quadratic equation.
Which method should we choose? If you need to review this topic, look back at SQE Target B. We don't see a way to factor the quadratic expression, so let's complete the square. |
\(x^2 - 6x - 48 = 0\)
\(x^2 - 6x - 48 + 48 = 0 + 48\) \(x^2 - 6x = 48\) \(x^2 - 6x + 9 = 48 + 9\) \((x - 3)^2 = 57\) \(x - 3 = \pm\sqrt{57}\) \(x = 3 \pm\sqrt{57}\) |
What is our solution or solutions? We need to look back at the context of the problem. The lengths of the sides of the rectangle must be positive because length is positive. Let's round our answer to the nearest tenth.
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\(x = 3 + \sqrt{57}\) or \(x = 3 - \sqrt{57}\)
\(x\approx 3 + 7.5\) or \(x\approx 3 - 7.5\) \(x\approx 10.5\) or \(x\approx -4.5\) \(x\approx 10.5\) because it will produce a positive length. |
Example 2:
You are building a rectangular brick patio surrounded by a flower border in a rectangular courtyard as shown. You would like to arrange the brick section in a \(12\) ft by \(17\) ft rectangle. The flower border is a consistent border of width \(x\) ft. Determine the width of your flower border if you have an area of \(300\) square feet available in your backyard. Solution: Watch the video for the solution. |
Quick Check
1) Your parents are building a deck of uniform width around your pool. The pool is \(18\) ft by \(10\) ft. They only have a total area of \(560\) ft. What is the widest the deck can be around the pool?
Quick Check Solution
1) Your parents are building a deck of uniform width around your pool. The pool is \(18\) ft by \(10\) ft. They only have a total area of \(560\) ft. What is the widest the deck can be around the pool?
Quick Check Solution