$2 x^2+2 x-4>\frac{1}{4}\left(x^2+4\right)$

Multiply both sides of the inequality by 4 and expand: x^2 + 4"> 8 x 2 + 8 x − 16 > x 2 + 4 .
Rearrange the terms to get a quadratic inequality: 0"> 7 x 2 + 8 x − 20 > 0 .
Use the quadratic formula to find the roots of 7 x 2 + 8 x − 20 = 0 : x 1 ​ ≈ − 2.356 and x 2 ​ ≈ 1.213 .
The solution to the inequality is x < − 2.356 or 1.213"> x > 1.213 , which can be written as x < 14 − 8 − 624 ​ ​ or \frac{-8 + \sqrt{624}}{14}"> x > 14 − 8 + 624 ​ ​ . \frac{-8 + \sqrt{624}}{14}}"> x < 14 − 8 − 624 ​ ​ or x > 14 − 8 + 624 ​ ​ ​

Explanation

Understanding the Problem We are given the inequality \frac{1}{4}\left(x^2+4\right)"> 2 x 2 + 2 x − 4 > 4 1 ​ ( x 2 + 4 ) . Our goal is to find the values of x that satisfy this inequality.

Eliminating the Fraction First, we multiply both sides of the inequality by 4 to eliminate the fraction: x^2+4"> 4 ( 2 x 2 + 2 x − 4 ) > x 2 + 4

Expanding the Inequality Next, we expand the left side of the inequality: x^2 + 4"> 8 x 2 + 8 x − 16 > x 2 + 4

Rearranging Terms Now, we move all terms to the left side to obtain a quadratic inequality: 0"> 8 x 2 − x 2 + 8 x − 16 − 4 > 0

Simplifying the Inequality We simplify the inequality: 0"> 7 x 2 + 8 x − 20 > 0

Using the Quadratic Formula To find the values of x that satisfy this inequality, we first find the roots of the quadratic equation 7 x 2 + 8 x − 20 = 0 . We use the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ where a = 7 , b = 8 , and c = − 20 .

Calculating the Discriminant We calculate the discriminant: D = b 2 − 4 a c = 8 2 − 4 ( 7 ) ( − 20 ) = 64 + 560 = 624

Finding the Roots Now, we find the roots: x 1 ​ = 14 − 8 − 624 ​ ​ ≈ 14 − 8 − 24.98 ​ ≈ − 2.356 and x 2 ​ = 14 − 8 + 624 ​ ​ ≈ 14 − 8 + 24.98 ​ ≈ 1.213

Determining the Solution Since the coefficient of x 2 is positive, the parabola opens upwards. The inequality 0"> 7 x 2 + 8 x − 20 > 0 is satisfied when x < x 1 ​ or x_2"> x > x 2 ​ . Therefore, the solution is x < 14 − 8 − 624 ​ ​ ≈ − 2.356 or \frac{-8 + \sqrt{624}}{14} \approx 1.213"> x > 14 − 8 + 624 ​ ​ ≈ 1.213


Examples
Understanding quadratic inequalities is crucial in various real-world applications. For instance, engineers use them to design stable structures, ensuring that stress and strain remain within safe limits. Economists apply them to model market behavior, predicting price fluctuations and optimizing investment strategies. In physics, quadratic inequalities help describe projectile motion, determining the range and maximum height of objects thrown into the air. These applications highlight the practical significance of mastering quadratic inequalities.

Answered by GinnyAnswer | 2025-07-05