Solve the equation: [tex]k^2-k-11=0[/tex]. Fully simplify [tex]k=[/tex]
Answer (2)
Apply the quadratic formula k = 2 a − b ± b 2 − 4 a c with a = 1 , b = − 1 , and c = − 11 .
Simplify the expression to k = 2 1 ± 45 .
Further simplify the square root to get k = 2 1 ± 3 5 .
The solutions are k = 2 1 + 3 5 and k = 2 1 − 3 5 , so the final answer is k = 2 1 ± 3 5 .
Explanation
Problem Analysis We are given the quadratic equation k 2 − k − 11 = 0 . Our goal is to find the values of k that satisfy this equation. We can use the quadratic formula to solve for k .
Quadratic Formula The quadratic formula is given by: k = 2 a − b ± b 2 − 4 a c where a , b , and c are the coefficients of the quadratic equation a k 2 + bk + c = 0 . In our case, a = 1 , b = − 1 , and c = − 11 .
Substitution Substitute the values of a , b , and c into the quadratic formula: k = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( − 11 )
Simplification Simplify the expression: k = 2 1 ± 1 + 44 k = 2 1 ± 45
Square Root Simplification Simplify the square root: 45 = 9 ⋅ 5 = 3 5
Final Solutions Therefore, the solutions are: k = 2 1 ± 3 5 The two solutions are: k = 2 1 + 3 5 ≈ 3.854 and k = 2 1 − 3 5 ≈ − 2.854
Conclusion Thus, the solutions to the equation k 2 − k − 11 = 0 are k = 2 1 + 3 5 and k = 2 1 − 3 5 .
Examples
Quadratic equations are used in various real-world applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its perimeter and area, and modeling growth and decay processes. For example, if you're launching a rocket, you can use a quadratic equation to model its path and predict where it will land, considering factors like initial velocity and launch angle. Understanding how to solve quadratic equations is essential for solving many practical problems in physics, engineering, and other fields.
The solutions to the equation k 2 − k − 11 = 0 using the quadratic formula are k = 2 1 + 3 5 and k = 2 1 − 3 5 .
;
