We shall now discuss problems that can be solved with quadratic equations.
To solve such problems, we first formulate the quadratic equation from the information we have, and then we solve it.
Consider these examples.
Example 1:
The product of two consecutive numbers is 72. Find the numbers.
Solution :
Let the smaller number = x
Let the larger number = x+1
Their product = x(x+1)
We are given that
x(x + 1) = 72
 x2 + x = 72
 x2 + x - 72 = 0
Factorizing
x2 + 9x - 8x - 72 = 0
x(x + 9) - 8(x + 9) = 0
(x - 8) (x + 9) = 0
 x - 8 = 0 or x + 9 = 0
 x= 8 or x = - 9
 x + 1 = 8 + 1 or x + 1 = -9 + 1
= 9 = -8
Therefore, the two consecutive numbers are 8, 9 or -9, -8
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Example 2:
A man walks a distance of 48 kilometers in a certain amount of time. If he increased his speed by 2 km/hr, he would have reached his destination four hours earlier. Find his usual speed.
Solution :
Let the man’s usual speed = x km/hr
Time taken to walk 48 km = (48/x) hrs
Time = Distance/Speed
Increased speed = (x + 2) km/hr
Time taken to walk 48 km =48/x+2
At a speed of (x + 2) km/hr, the man reaches his destination 4 hours earlier.
 96 = 4(x2 + 2x)
96 = 4x2+ 8x
Transposing
4x2 + 8x - 96 = 0
 4(x2 + 2x - 96) = 0
 x2 + 2x - 24 = 0/4
 x 2+ 2x - 24 = 0
Factorizing
x2 + 6x - 4x - 24 = 0
x(x + 6) - 4 (x+6) = 0
 (x - 4) (x + 6) = 0
 x - 4 = 0 or x + 6 = 0
 x = 4 or x = -6.
Since the man cannot walk at a negative
pace, we reject x = -6
Speed = x =4km/hr
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