Finding the Roots of a Quadratic Equation Examples :
In this section, we will learn, how to find the roots of a quadratic equation.
Example 1 :
Find the roots of the following equations:
x - (1/x) = 3, x ≠ 0
Solution :
(x² - 1)/x = 3
x² - 1 = 3 x
x² - 3 x - 1 = 0
a = 1, b = -3, c = -1
x = - b ± √(b² - 4 a c)/2a
x = - (-3) ± √((-3)² - 4 (1)(-1))/2(1)
x = 3 ± √(9+4)/2
x = (3 ± √13)/2
Example 2 :
Find the roots of the following equations:
[1/(x + 4)] - [1/(x -7)] = 11/30
Solution :
[1/(x + 4)] - [1/(x -7)] = 11/30
[(x - 7) - (x + 4)]/(x + 4)(x - 7) = 11/30
(x - 7 - x - 4)/(x² + 4 x - 7 x - 28) = 11/30
-11/(x² - 3 x - 28) = 11/30
30 (-11) = 11 (x² - 3 x - 28)
-30 = x² - 3 x - 28
x² - 3 x - 28 + 30 = 0
x² - 3 x + 2 = 0
x² - 1 x - 2 x + 2 = 0
x (x - 1) - 2 (x - 1) = 0
(x - 2) (x - 1) = 0
x - 2 = 0 x - 1 = 0
x = 2 x = 1
Example 3 :
The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is 1/3. Find his present age.
Solution :
Let 'x' be Rehman's present age
Three years ago his age was x - 3
5 years after his age will be x + 5
Sum of the reciprocals of Rehman's ages = 1/3
[1/(x -3) + 1/(x + 5)] = 1/3
(x + 5 + x - 3)/(x-3)(x + 5) = 1/3
(2 x + 2)/(x² - 3 x + 5 x - 15) = 1/3
(2 x + 2)/(x² + 2 x - 15) = 1/3
3 (2 x + 2) = 1 (x² + 2 x - 15)
6 x + 6 = x² + 2 x - 15
x² + 2 x - 15 - 6 x - 6 = 0
x²- 4 x - 21 = 0
x² - 7 x + 3 x - 21 = 0
x (x - 7) + 3 (x - 7) = 0
(x + 3) (x - 7) = 0
x = 3 or x = 7
So his present age will be 7 years.
Example 4 :
In a class test,the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
Solution :
Let 'x' be the mark got by Shefali in the subject mathematics
let '30 - x' be the mark got by Shefali in English
(x + 2)(30 - 3 - x) = 210
(x + 2)(27 - x) = 210
27 x - x² + 54 - 2 x - 210 = 0
- x² + 25 x - 156 = 0
x² - 25 x + 156 = 0
x² - 12 x - 13 x + 156 = 0
x (x - 12) - 13 (x - 12) = 0
(x - 12) (x - 13) = 0
x - 12 = 0 x - 13 = 0
x = 12 x = 13
If Shefali's mathematics marks = 12 then her English mark will be = 30 - 12 = 18.
If Shefali's mathematics marks = 13 then her English mark will be = 30 - 13 = 17.
After having gone through the stuff given above, we hope that the students would have understood, finding the roots of a quadratic equation examples.
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