Applications using Factoring

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Set up an algebraic equation and then solve the following problems.
An integer is 3 less than 5 times another. If the product of the two integers is 36, then find the integers.
The width of a rectangle is 5 units less than the length. If the area is 150 square units, then find the dimensions of the rectangle.
The length of a rectangle is 4 inches more than its width. The area of the rectangle is equal to 5 inches more than 2 times the perimeter. Find the length and width of the rectangle.
The height of a projectile launched upward at a speed of 32 feet/second from a height of 128 feet is given by the function h(t)=-16t^2+32t+128. How long will it take the projectile to hit the ground?
The height of an object dropped from the top of a 144-foot building is given by h(t)=-16t^2+144. How long will it take the object to hit the ground?

Sample Solution

Here are the solutions to the problems, along with the algebraic equations used:

1. Integer Problem:

  • Equations:

    • Let x be the first integer.
    • The second integer is 5x – 3.
    • Their product: x(5x – 3) = 36

  • Solution:

    • 5x^2 – 3x – 36 = 0
    • Factoring: (5x + 9)(x – 4) = 0
    • x = 4 or x = -9/5
    • The integers are 4 and 17 (discarding the negative value).

2. Rectangle Dimensions:

  • Equations:

    • Let l be the length.
    • The width is l – 5.
    • Area: l(l – 5) = 150

  • Solution:

    • l^2 – 5l – 150 = 0
    • Factoring: (l – 15)(l + 10) = 0
    • l = 15 or l = -10 (discarding the negative value).
    • The dimensions are 15 units by 10 units.

3. Rectangle with Area and Perimeter:

  • Equations:

    • Let w be the width.
    • The length is w + 4.
    • Area: w(w + 4)
    • Perimeter: 2(2w + 4)
    • Given: w(w + 4) = 2(2w + 4) + 5

  • Solution:

    • w^2 + 4w = 4w + 13
    • w^2 = 13
    • w = √13 (approximately 3.61 inches)
    • The length is 7.61 inches.

4. Projectile Height:

  • Equation:

    • h(t) = -16t^2 + 32t + 128
    • The projectile hits the ground when h(t) = 0.

  • Solution:

    • -16t^2 + 32t + 128 = 0
    • Divide by -4: 4t^2 – 8t – 32 = 0
    • Factoring: (2t – 8)(2t + 4) = 0
    • t = 4 or t = -2 (discarding the negative value).
    • It takes 4 seconds to hit the ground.

5. Dropped Object Height:

  • Equation:

    • h(t) = -16t^2 + 144
    • The object hits the ground when h(t) = 0.

  • Solution:

    • -16t^2 + 144 = 0
    • Divide by -16: t^2 – 9 = 0
    • Factoring: (t – 3)(t + 3) = 0
    • t = 3 or t = -3 (discarding the negative value).
    • It takes 3 seconds to hit the ground.

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