Quadratic Inequations

Quadratic Inequations

If ax² + bx + c is a quadratic expression, then ax² + bx + c > 0 (or) ax² bx + c ≥ 0 (or) ax² bx + c < 0 (or) ax² bx + c ≤ 0 is called a quadratic inequation (or) quadratic inequality.

Examples:

(i) 3x² – 5x + 7 > 0,

(ii) 2x² + 3x – 5 ≥ 0,

(iii) x² – 2x – 3 < 0,

(iv) 4x² + 5x – 1 ≤ 0.

Example: if x is real, then show that \(\frac{{{x}^{2}}+34x-71}{{{x}^{2}}+2x-7}\) does not lie between 5 and 9.

Solution: Given that \(\frac{{{x}^{2}}+34x-71}{{{x}^{2}}+2x-7}\),

Let us consider \(y=\frac{{{x}^{2}}+34x-71}{{{x}^{2}}+2x-7}\),

yx² + 2xy – 7y = x² + 34x – 71

⇒ (y – 1) x² + (2y – 34) x + (71 – 7y) = 0

x ϵ R

⇒ D = b² – 4ac ≥ 0

⇒ (2y – 34)² – 4 (y – 1) (71 – 7y) ≥ 0

⇒ 4y² + 1156 – 136y – 4 (71y – 7y² – 71 + 7y) ≥ 0

⇒ 32y² – 448y + 1440 ≥ 0

⇒ 32y² – 448y + 1440 = 0

⇒ y² – 14y + 45 = 0

⇒ (y – 5) (y – 9) = 0

⇒ y = 5, 9

⇒ 32y² – 448y + 1440 ≥ 0

⇒ y ≤ 5 (or) y ≥ 9

⇒ y does not lie between 5 and 9.