Find the roots of the following quadratic equations, if they exist by the method of completing the square:
9x2 – 15x + 6 = 0
Answers
Question:
Find the roots of the following quadratic equations, if they exist by the method of completing the square: 9x² - 15x + 6 = 0
Answer:
x = 1 , 2/3
Note:
• The possible values of unknown (variable) for which the equation is satisfied are called its solutions or roots .
• If x = a is a solution of any equation in x , then it must satisfy the given equation otherwise it's not a solution (root) of the equation.
• The discriminant of the the quadratic equation
ax² + bx + c = 0 , is given as ; D = b² - 4ac
• If D > 0 then its roots are real and distinct.
• If D < 0 then its roots are imaginary.
• If D = 0 then its roots are real and equal.
Solution:
Here,
The given quadratic equation is :
9x² - 15x + 6 = 0
Clearly, here we have ;
a = 9
b = -15
c = 6
Now,
The discriminant will be ;
=> D = b² - 4ac
=> D = (-15)² - 4•9•6
=> D = 225 - 216
=> D = 9. ( D > 0 )
Since,
The discriminant of the given quadratic equation is greater than zero, thus there must exist two distinct real roots.
Now,
=> 9x² - 15x + 6 = 0
Dividing both sides by 9 , we have ;
=> x² - 15x/9 + 6/9 = 0
=> x² - 5x/3 + 2/3 = 0
=> x² - 5x/3 + (5/6)² - (5/6)² + 2/3 = 0
=> x² - 5x/3 + (5/6)² = (5/6)² - 2/3
=> x² - 2•x•(5/6) + (5/6)² = 25/36 - 2/3
=> (x - 5/6)² = (25 - 24)/36
=> (x - 5/6)² = 1/36
=> x - 5/6 = √(1/36)
=> x - 5/6 = ± 1/6
=> x = 5/6 ± 1/6
=> x = (5±1)/6
=> x = (5+1)/6 , (5-1)/6
=> x = 6/6 , 4/6
=> x = 1 , 2/3
Hence,
The roots of the given quadratic equation are :
x = 1 , 2/3 .