We can follow the steps given below to solve word problems using quadratic equations.
Step 1 :
Understanding the question is more important than any other thing. That is, always it is very important to understand the information given in the question rather than solving.
Step 2 :
If it is possible, we have to split the given information. Because, when we split the given information in to parts, we can understand them easily.
Step 3 :
Once we understand the given information clearly, solving the word problem in quadratic equation would not be a challenging work.
Step 4 :
When we try to solve the word problems in quadratic equations, we have to introduce "x" or some other alphabet for unknown value (=answer for our question) and form a quadratic equation with this "x". Finally we have to get value for the alphabet which was introduced for the unknown value.
Step 5 :
If it is required, we have to draw picture for the given information. Drawing picture for the given information will give us a clear understanding about the question.
Step 6 :
Using the alphabet introduced for unknown value, we have to translate the English statement (information) given in the question as quadratic equation equation.
In translation, we have to translate the following English words as the corresponding mathematical symbols.
of -----> x (multiplication)
am, is, are, was, were, will be, would be --------> = (equal)
Step 7 :
Once we have translated the English Statement (information) given in the question as quadratic equation correctly, 90% of the work will be over. The remaining 10% is just getting the answer. That is solving for the unknown.
These are the steps most commonly involved in solving word problems in quadratic equations.
Let us see how the above explained steps work in solving word problems using quadratic equations.
Problem :
A piece of iron rod cost $ 60. If the rod was 2 meter shorter and each meter costs $ 1 more and the total cost would remain unchanged. What is the length of the rod?
Solution:
Step 1 :
Let us understand the given information. There are three information given in the question.
1. A piece of iron rod costs $ 60.
2. If the rod was 2 meter shorter and each meter costs $ 1 more
3. Total cost would remain unchanged.
Step 2 :
Target of the question : What is the length of the rod?
Step 3 :
Let "x" be the length of the rod.
Clearly, we have to find the value of "x"
Step 4 :
If the rod is 2 meter shorter, length of the rod is
= (x-2)
Step 5 :
From the third information, we have the following statements.
Total cost of rod having length x meters is $ 60.
Total cost of rod having length (x-2) meters is $ 60.
Step 6 :
Cost of 1 meter of rod having length x meters is
= 60 / x -----(1)
Cost of 1 meter of rod having length (x-2) meters is
= 60 / (x - 2) -----(2)
Step 7 :
From the second information, we can consider the following example.
That is, if the cost of 1 meter of rod x is $10, then the cost of 1 meter of rod (x-2) will be $11.
$10 & $11 can be balanced as shown below.
10 + 1 = 11
(This is just for en example)
Step 8 :
If we apply the same logic for (1) & (2), we get
(60 / x) + 1 = 60 / (x - 2)
(60 + x) / x = 60 / (x - 2)
(x + 60)(x - 2) = 60x
x^{2} + 58x - 120 = 60x
x^{2} - 2x - 120 = 0
(x - 12)(x + 10) = 0
x - 12 = 0 or x + 10 = 0
x = 12 or x = -10
Because length can never be a negative value, we can ignore x = -10.
Therefore,
x = 12
So, the length of the rod is 12 meter.
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