Carmen planted daylilies on a square piece of land with an area of 49 square feet. She wants to plant petunias around the daylilies to form a border whose area is 72 square feet. What length should she make the sides of the outer square?
Let x represent the length of
a side of the outer square. Find a relationship between the variable and given
numerical information to write and solve an equation.
Length cannot be negative, so the sides of the outer square
measure 11 feet.
We were able to solve the problem
easily because 121 is a perfect square. However, few quadratic expressions are
perfect squares. To make any quadratic expression a perfect square, a
method called completing
the square can be used.
When given an equation, you can
complete the square by writing one side of the equation as a perfect square.
This is modeled below.
To complete the square for any
quadratic expression of the form x2 + bx, you can
follow the steps below.
1.
Find the value of c that makes x2 + 14x +
c a perfect square.
Alternative Solutions:
Thus, c
= 49. Notice that x2 + 14x + 49 = (x + 7)2.
Once a perfect square is found, we can
solve the equation by taking the square root of each side.
Example
2.
Solve x2 + 12x – 13 = 0 by completing the square.
Alternative Solutions:
The solutions are –13 and
1. Check the solution by
using the graphing calculator
or by substituting 13 and 1 for x in the original equation.
You can complete the square only if
the coefficient of the first term is 1. If the coefficient is not 1,
first divide each term by the coefficient.
Construction
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3. A school wants to redesign its nurse’s station. Because of building codes,
the maximum length of the rectangular room is 20 meters less than twice its width.
Find the dimensions, to the nearest tenth of a meter, for the widest possible
nurse’s station if its area is to be 60 square meters.
Alternative Solutions:
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