Solving quadratic equations by factoring word problems

Problem 1 :

Two natural numbers differ by 2 and their product is 360. Find the numbers.

Solution :

Let x and y be two natural numbers

it differs by 2

So,

x - y  =  2 -----(1)

Their product is 360

So,

xy  =  360

y  =  360/x ----- (2)

Now we may apply the value of y in the first equation.

x - (360/x)  =  2

(x2 - 360)/x  =  2

x2 - 360  =  2x

x2 - 2x - 360  =  0

(x + 18)(x - 20)  =  0

Because it is a positive integer, we should not take x = -18.  We take positive value for x.

If x  =  20,

then

y  =  360/20

y  =  18

Therefore the required positive integers are 20 and 18.

Verification :

Two natural numbers differ by 2.

20 - 18  =  2

their product is 360

20(18)  =  360

Problem 2 :

There are three consecutive positive integers such that the sum of the square of first and the product of the other two is 154. Find the integers.

Solution :

Let x, (x + 1) and (x + 2) be the first three consecutive integers

The sum of the squares of first and the product of the other two is 154

x2 + (x + 1)(x + 2)  =  154

x2 + x2 + 2x + 1x + 2  =  154

2x2 + 3x + 2  =  154

2x2 + 3x + 2 - 154  =  0

2x2 + 3x - 152  =  0

(2x + 19)(x - 8)  =  0

Since it is positive integer, the value of x be 8.

Therefore three consecutive integers are 8, 9 and 10.

Verification :

The sum of the square of first and the product of the other two is 154.

82 + (9)(10)  =  154

 64 + 90  =  154

 154  =  154

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Quadratic equations - Solving word problems using factoring of trinomials
Question 1a:
Find two consecutive integers that have a product of 42

Quadratic equations - Solving word problems using factoring of trinomials
Question 1b:
There are three consecutive integers. The product of the two larger integers is 30. Find the three integers.

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Quadratic Equations - Solving Word problems by Factoring
Question 1c:
A rectangular building is to be placed on a lot that measures 30 m by 40 m. The building must be placed in the lot so that the width of the lawn is the same on all four sides of the building. Local restrictions state that the building cannot occupy any more than 50% of the property. What are the dimensions of the largest building that can be built on the property?

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More Word Problems Using Quadratic Equations
Example 1
Suppose the area of a rectangle is 114.4 m2 and the length is 14 m longer than the width. Find the length and width of the rectangle.

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More Word Problems Using Quadratic Equations
Example 2
A manufacturer develops a formula to determine the demand for its product depending on the price in dollars. The formula is
D = 2,000 + 100P - 6P2
where P is the price per unit, and D is the number of units in demand. At what price will the demand drop to 1000 units?

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More Word Problems Using Quadratic Equations
Example 3
The length of a car’s skid mark in feet as a function of the car’s speed in miles per hour is given by
l(s) = .046s2 - .199s + 0.264
If the length of skid mark is 220 ft, find the speed in miles per hour the car was traveling.

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