Mid-Chapter Quiz
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Express the length and width of the new slab in terms of x.
We need the positive solution to the quadratic equation found in Part A.
x^2+14x-72=0
Length: 12 feet
Width: 10 feet
We are given the dimensions of the original rectangular concrete slab and are asked to find the dimensions of the new one. It is given that the length and width of the new slab are increased by the same amount.
New Length:& 8+xfeet
New Width:& 6+xfeet
The area of a rectangle is the product if its length and width.
Distribute (6+x)
Distribute 8 & x
Add terms
Commutative Property of Addition
LHS-120=RHS-120
We found a quadratic equation in standard form that represents the area of the new slab. x^2+14x-72=0
To find the dimensions of the new slab, let's solve the equation we have found in Part A first. The solution will tell us the amount of increase in the sides.
We can use the method of factoring to solve our equation. To factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
x^2+14x -72
In this case, we have -72. This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have the opposite signs, one positive and one negative.
| Factor Constants | Product of Constants | |
|---|---|---|
| 1 and -72 | -1 and 72 | -72 |
| 2 and -36 | -2 and 36 | -72 |
| 3 and -24 | -3 and 24 | -72 |
| 4 and -18 | -4 and 18 | -72 |
| 6 and -12 | -6 and 12 | -72 |
| 8 and -9 | -8 and 9 | -72 |
Next, let's consider the coefficient of the linear term x. x^2+ 14x -72 For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 14. Since this coefficient is positive, we will only check the factors where the sum is positive.
| Factors | Sum of Factors |
|---|---|
| -1 and 72 | 71 |
| -2 and 36 | 34 |
| -3 and 24 | 21 |
| -4 and 18 | 14 |
| -6 and 12 | 6 |
| -8 and 9 | 1 |
We found two numbers whose product is -72 and whose sum is 14. These two numbers can be used to write the factored form. x^2+ 14x -72=(x-4)(x+18) To solve this equation, we will apply the Zero Product Property.
Use the Zero Product Property
(I): LHS+4=RHS+4
(II): LHS-18=RHS-18
Since x represents the amount of increase in the sides, only the positive solution is meaningful. x=4 We can add this value to the dimensions of the original slab to find the dimensions of the new slab. New Length:& 8+4=12feet New Width:& 6+4=10feet To quickly check, we can multiply these dimensions to see that the area is indeed 10* 12=120 square feet.