Question

Find the roots of the following quadratic equations, if they exist by the method of completing the square:
9x2 – 15x + 6 = 0

Answer 1

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 .