Question

By using the method of completing the square, show that the equation 2x^2 + x + 4 = 0 has no real roots.

Answer 1

2x² + x + 4 = 0

(Divide by 2 through)

x² + 1/2x + 2 = 0

(Subtract 2 from both sides)

x² + 1/2x =  -2

(add (b/2)² to both sides)

x² + 1/2x  +  (1/4)² =  -2  + (1/4)²

(Complete the square)

(x + 1/4)² = -31/16

(Square root both sides)

x + 1/4 = ±√(-31/16)

x + 1/4 = ±i√(31/16)

Subtract 1/4 from both sides:

x = i√(31/16) - 1/4 or - i√(31/16) - 1/4

=> Both roots are imaginary numbers

=> There is no real roots

Answer 2

Hey there !!


▶ The given quadratic equation :-

°•° 2x² + x + 4 = 0.

[ Multiplying both side by 2, we get ] .

=> 4x² + 2x + 8 = 0.

=> 4x² + 2x = -8 .

[ Adding ( ½ )² on both side ].

=> 4x² + 2x + ( ½ )² = -8 + ( ½ )² .

$= > {(2x)}^{2} + 2 \times 2x \times \frac{1}{2} + {( \frac{1}{2} )}^{2} = - 8 + {( \frac{1}{2} )}^{2} . \\ \\ = > {(2x + \frac{1}{2} )}^{2} = - 8 + \frac{1}{4} . \\ \\ = > {(2x + \frac{1}{2} )}^{2} = \frac{ - 32 + 1}{4} . \\ \\ = > {(2x + \frac{1}{2} )}^{2} = \frac{ - 31}{4} < 0.$


But , ${(2x + \frac{1}{2} )}^{2}$ cannot be negative for any real value of x .

So, there is no real value of x that satisfies the given equation.


✔✔ Hence , the given equation has no real roots ✅✅.



THANKS


#BeBrainly.