By using the method of completing the square, show that the equation 2

1.	By using the method of completing the square, show that the equation 2x2 + x + 4 = 0 has no real roots
Answer» 2x² + x+ 4 = 0 ⇒ 4x² +2x + 8 = 0 (Multiplying both sides by 2) ⇒ 4x² +2x = - 8 ⇒ (2x)² + 2 \(\times\) 2x \(\times\) 1/2 + (1/2)2 = - 8 + (1/2)² [Adding (1/2)² on both sides] ⇒ (2x + 1/2)² = - 8 + 1/4 = - 31/4 < 0 But, (2x + 1/2)² cannot be negative for any real value of x So, there is no real value of x satisfying the given equation. Hence, the given equation has no real roots.

By using the method of completing the square, show that the equation 2x2 + x + 4 = 0 has no real roots

Answer»

2x² + x+ 4 = 0

⇒ 4x² +2x + 8 = 0

(Multiplying both sides by 2)

⇒ 4x² +2x = - 8

⇒ (2x)² + 2 \(\times\) 2x \(\times\) 1/2 + (1/2)2 = - 8 + (1/2)²

[Adding (1/2)² on both sides]

⇒ (2x + 1/2)² = - 8 + 1/4 = - 31/4 < 0

But, (2x + 1/2)² cannot be negative for any real value of x

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

Hence, the given equation has no real roots.