How to use quadratic equations to solve for the sides of a rectangle

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How to use quadratic equations to solve for the sides of a rectangle
Although not essential, a calculator will make finding the solution easier. (Jupiterimages/Polka Dot/Getty Images)

If you are given the area of a rectangle, as well as some information about its perimeter or sides, you can derive a simple quadratic equation based on the rectangle's geometric properties. You can then solve this quadratic equation to obtain the rectangle's two sides. The exact process varies depending on whether you are given the rectangle's perimeter, the difference between its length and width, or its length and width expressed as a linear function of x.

Skill level:
Moderately Easy

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Instructions

    Given the length and width as a linear function of x

  1. 1

    Call the rectangle's length "l," its width "w." and its area "a." Because the area of a rectangle is equal to its length multiplied by its width, you can create a quadratic equation using the information you are given. For example, if:

    l = 2x + 4 w = 2x + 1 a = 40

    Then:

    (2x + 4) * (2x + 1) = 40 4x^2 + 8x + 2x + 4 = 40 4x^2 + 10x - 36 = 0 2x^2 + 5x - 18 = 0

  2. 2

    Solve the equation for x. For example:

    x = (-5 +/- sqrt(25 +144)) / 4 x = (-5 +/- 13) / 4

    x = 8 / 4 OR x = -18 / 4 x = 2 OR x = -4.5

  3. 3

    Insert the value of x in the original formulas you were given to obtain the length and width of the rectangle. For example:

    For x = 2, l = (2 * 2) + 4 = 8 and w = (2 * 2) +1 = 5. For x = -4.5, l = (2 * -4.5) + 4 = -5 and w = (2 * -4.5) + 1 = -8.

    If a particular value of x results in a negative value for l or w, as in this case, discard that solution. Multiply the resulting length by the resulting width to check your work.

    Given either the perimeter or the difference between length and width

  1. 1

    Call the two sides of the rectangle "x" and "y." If you have been given the value of y in terms of x, skip to the next step. If you have been given the rectangle's perimeter, use the fact that the perimeter of a rectangle is twice the sum of its sides to derive the value of y in terms of x. For example, assuming a rectangle with a perimeter of 26:

    2 * (x + y) = 26 x + y = 13 y = 13 - x

  2. 2

    Replace y with its value in terms of x in the rectangle area formula to obtain the quadratic equation you need to solve. For example, assuming the rectangle in question has an area of 40:

    x * y = 40 x * (13 - x) = 40 13x - x^2 = 40 13x - x^2 - 40 = 0 x^2 - 13x + 40 = 0

  3. 3

    Solve the equation for x. The two solutions correspond to the two sides of the rectangle. For example:

    x = (13 +/- sqrt(13^2 - 4 * 40)) / 2 x = (13 +/- sqrt(169 - 160)) / 2 x = (13 +/- sqrt(9)) / 2 x = (13 +/- 3) / 2

    x = (13 - 3) / 2 OR x = (13 + 3) / 2 x = 10 / 2 OR x = 16 / 2 x = 5 OR x = 8

    Calculate the rectangle's perimeter and area using the two solutions and compare the results to the values you were given to check your work.

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