Solve 2x2 + 5root3 + 6 = 0 by the completing square method
Answers
Answer:
Roots of the equation are - √3/2 and - 2√3.
Step-by-step explanation:
Looks like the equation is 2x² + 5√3x + 6 = 0
Given Equation :
⇒ 2x² + 5√3x + 6 = 0
Dividing on both sides by 2
⇒ 2x²/2 + 5√3/2 + 6/2 = 0
⇒ x² + 5√3/2 + 3 = 0
Transposing 3 to RHS
⇒ x² + 5√3/2 = -3
It can be written as
⇒ x² + 2( x )( 5√3/4 ) = - 3
Adding ( 5√3/4 )² on both sides
⇒ x² + 2( x )( 5√3/4 ) + ( 5√3/4 )² = - 3 + ( 5√3/4 )²
Since ( a + b )² = a² + 2ab + b²
⇒ ( x + 5√3/4 )² = - 3 + 75/16
⇒ ( x + 5√3/4 )² = ( - 48 + 75 ) / 16
⇒ ( x + 5√3/4 )² = 27/16
Taking square root on both sides
⇒ x + 5√3/4 = ± √27/16
⇒ x + 5√3/4 = ± √27 / √16
⇒ x + 5√3/4 = ± 3√3/4
⇒ x + 5√3/4 = 3√3/4 OR x + 5√3/4 = - 3√3/4
⇒ x = 3√3/4 - 5√3/4 OR x = - 3√3/4 - 5√3/4
⇒ x = - 2√3/4 OR x = - 8√3/4
⇒ x = - √3/2 OR x = - 2√3
Hence - √3/2 and - 2√3 are the roots of the equation.
x = -√3/2 or - 2√3
Step-by-step explanation:
2x² + 5√3x + 6 = 0
Dividing both sides by 2, we get: x² + 5√3/2 x + 3 = 0
x² + 5√3/2 x = -3
Let's add (5√3/4 )² on both sides of the equation.
x² + 2[5√3/4]x + (5√3/4 )² = -3 + (5√3/4 )²
From the expression (a + b)² = a² + 2ab + b², we can rewrite the above as:
( x + 5√3/4 )² = -3 + 25*3/16
( x + 5√3/4 )² = (-48 + 75) / 16
( x + 5√3/4 )² = 27/16
Therefore x + 5√3/4 )² = √27/16
x + 5√3/4 = ± 3√3/4
When x + 5√3/4 = +3√3/4, x = 3√3/4 - 5√3/4
Therefore x = -√3/2.
When x + 5√3/4 = -3√3/4, x = -3√3/4 - 5√3/4
Therefore x = -2√3
Hence solved. x = -√3/2 or - 2√3