Question

Find the roots of each of the following quadratic equations if they exist by the method of completing the squares:
2x2 + √15 x + √2 = 0

Answer 1

Question:

Find the roots of the following quadratic equations, if they exist by the method of completing the square: 2x² + √15x + √2 = 0

Answer:

x = - √15/4 ± √(15-8√2)/4

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 :

2x² + √15x + √2 = 0

Clearly, here we have ;

a = 2

b = √15

c = √2

Now,

The discriminant will be ;

=> D = b² - 4ac

=> D = (√15)² - 4•2•√2

=> D = 15 - 8√2 ( D > 0 )

Since,

The discriminant of the given quadratic equation is greater than zero, thus there must exist two distinct real roots .

Now,

=> 2x² + √15x + √2 = 0

Dividing both sides by 2 , we have ;

=> x² + √15x/2 + √2/2 = 0

=> x² + √15x/2 + (√15/4)² - (√15/4)² + √2/2 = 0

=> x² + √15x/2 + (√15/4)² = (√15/4)² - √2/2

=> x² + 2•x•(√15/4) + (√15/4)² = 15/16 - √2/2

=> (x + √15/4)² = (15-8√2)/16

=> x + √15/4 = √[(15-8√2)/16]

=> x + √15/4 = ± √(15-8√2)/4

=> x = - √15/4 ± √(15-8√2)/4

Hence,

The roots of the given quadratic equation are:

x = - √15/4 ± √(15-8√2)/4