Which equation fits a parabola with an a-value of 1, a b-value of 3, and a c-value of 6?
$0=2 x-5+x^4$
$0=x-3-5 x^2$
$0=3 x-5-x^2$
$0=3 x+5 x^2$
Answer (1)
None of the provided equations have 1, 3, and 6 as solutions.
Explanation
Problem Analysis The question asks which of the given equations has solutions of 1, 3, and 6. We will test each equation to see if these values are indeed solutions.
Testing Equation 1 Equation 1: 0 = 2 x − 5 + x 4 . We substitute x = 1 : 2 ( 1 ) − 5 + ( 1 ) 4 = 2 − 5 + 1 = − 2 = 0 . Since 1 is not a solution, we don't need to check 3 and 6.
Testing Equation 2 Equation 2: 0 = x − 3 − 5 x 2 . We substitute x = 1 : 1 − 3 − 5 ( 1 ) 2 = 1 − 3 − 5 = − 7 = 0 . Since 1 is not a solution, we don't need to check 3 and 6.
Testing Equation 3 Equation 3: 0 = 3 x − 5 − x 2 . We substitute x = 1 : 3 ( 1 ) − 5 − ( 1 ) 2 = 3 − 5 − 1 = − 3 = 0 . Since 1 is not a solution, we don't need to check 3 and 6.
Testing Equation 4 Equation 4: 0 = 3 x + 5 x 2 . We substitute x = 1 : 3 ( 1 ) + 5 ( 1 ) 2 = 3 + 5 = 8 = 0 . Since 1 is not a solution, we don't need to check 3 and 6.
Conclusion None of the given equations have 1, 3, and 6 as solutions. Therefore, none of the equations satisfy the condition.
Examples
Understanding the solutions to equations is crucial in many fields. For example, in physics, finding the roots of an equation can help determine equilibrium points in a system. In engineering, solutions to equations can represent stable states of a structure. In economics, they can represent market equilibrium. By understanding how to find solutions, we can model and analyze real-world phenomena.
