Given g(x)=−x2+10x−3, find the following.
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Answer:
Total area equation = tex]w(w+6)=91[/tex]
b/2 = 3
Dimensions of the patio: width = 7 feet, length = 13 feet
Step-by-step explanation:
The area of a rectangle is given the formula:
where
is the width
is the length
We know from our problem that the area of the patio is 91 square feet, so . We also know that the length is 6 feet longer then the width, so
.
Replacing values in our area equation
Expanding the left side:
Remember that to complete the square we need to add half the coefficient of the linear term squared. The lineal term is , so its coefficient is 6. Now, half its coefficient or
. Finally,
.
To complete the square we need to add 9 to both sides of the equation:
Notice that the left side is a perfect square trinomial (both and 9 are perfect squares), so we can express it as:
Now that we completed the square, we can solve our equation
- Take square root to both sides
- Subtract 3 from both results
Since length cannot be negative, is the solution of our equation.
We now know that the width of our rectangular patio is 7 feet, so we can find its length:
We can conclude that half the coefficient of the width is , the width of the patio is 7 feet, and its length is 13 feet.