$2 \omega^2+2 \omega-11=0$

Identify the quadratic equation: 2 ω 2 + 2 ω − 11 = 0 .
Apply the quadratic formula: ω = 2 a − b ± b 2 − 4 a c ​ ​ , where a = 2 , b = 2 , c = − 11 .
Substitute the values into the formula and simplify: ω = 2 ( 2 ) − 2 ± 2 2 − 4 ( 2 ) ( − 11 ) ​ ​ = 2 − 1 ± 23 ​ ​ .
State the final answer: ω = 2 − 1 ± 23 ​ ​ ​ .

Explanation

Understanding the Problem We are given the quadratic equation 2 ω 2 + 2 ω − 11 = 0 and we want to find the values of ω that satisfy this equation.

Applying the Quadratic Formula To solve the quadratic equation, we can use the quadratic formula, which is given by: ω = 2 a − b ± b 2 − 4 a c ​ ​ where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 . In our case, a = 2 , b = 2 , and c = − 11 .

Substituting the Values Now, we substitute the values of a , b , and c into the quadratic formula: ω = 2 ( 2 ) − 2 ± 2 2 − 4 ( 2 ) ( − 11 ) ​ ​ ω = 4 − 2 ± 4 + 88 ​ ​ \omega = \frac{-2 \pm \sqrt{92}}{4}$ 4. Simplifying the Expression We can simplify the square root of 92 as $\sqrt{92} = \sqrt{4 \times 23} = 2\sqrt{23}$. Therefore, \omega = \frac{-2 \pm 2\sqrt{23}}{4} \omega = \frac{-1 \pm \sqrt{23}}{2} So the two solutions for $\omega$ are: \omega_1 = \frac{-1 + \sqrt{23}}{2} \omega_2 = \frac{-1 - \sqrt{23}}{2} W ec ana pp ro x ima t e t h ese v a l u es a s : \omega_1 \approx \frac{-1 + 4.7958}{2} \approx 1.8979 \omega_2 \approx \frac{-1 - 4.7958}{2} \approx -2.8979 . 5. Final Answer Therefore, the solutions to the quadratic equation $2 \omega^2+2 \omega-11=0$ are: \omega = \frac{-1 + \sqrt{23}}{2} \approx 1.8979 \omega = \frac{-1 - \sqrt{23}}{2} \approx -2.8979 Thus, the solutions are ω = 2 − 1 ± 23 ​ ​ ​ .


Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its perimeter and area, or modeling the growth of a population. For example, if you want to launch a rocket and need to know its path, you can use a quadratic equation to model its trajectory, considering factors like initial velocity and gravity. Understanding how to solve quadratic equations is essential for making accurate predictions and informed decisions in these situations.

Answered by GinnyAnswer | 2025-07-08