Quadratic Equation Calculus Proof

In summary, by calculating the minimum value of a general quadratic function, it can be shown that the function will always be greater than or equal to zero for all values of x if and only if b2-4ac is less than or equal to zero. This can be found by taking the derivative of the function and setting it equal to zero, which leads to a simplified form of c+ax^2.
  • #1
harrietstowe
46
0

Homework Statement


Consider the general quadratic function f(x)=ax2+bx+c with a>0. By calculating the minimum value of this function, show that f(x)[tex]\geq[/tex]0 for all x if and only if b2-4ac[tex]\leq[/tex]0


Homework Equations





The Attempt at a Solution


I know that the min for a quadratic equation is at x=-b/2a and I proved this by taking the derivative of f(x) and setting it equal to zero. I then plugged this in for x to get c+ax2 but I don't know where to go from there.
Thank You
 
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  • #2
You plugged in x=-b/2a wrong... I mean, your formula still shows an x. That shouldn't be there...
 
  • #3
Ahh now I see. Thank you
 

1. What is a quadratic equation?

A quadratic equation is a mathematical expression that contains a variable, usually represented by x, and includes at least one squared term. It can be written in the form ax² + bx + c = 0, where a, b, and c are constants.

2. How do you solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula, which is x = (-b ± √(b² - 4ac)) / 2a. Alternatively, you can factor the equation and set each factor equal to zero, or complete the square to find the solutions.

3. What is the importance of calculus in quadratic equations?

Calculus is important in quadratic equations because it allows us to find the critical points, or turning points, of the graph. These points represent the maximum or minimum values of the quadratic function, which can be useful in real-world applications.

4. How do you prove that a quadratic equation has no real solutions?

A quadratic equation has no real solutions if the discriminant, b² - 4ac, is negative. This can be proven using the quadratic formula or by analyzing the graph of the equation, which will not intersect with the x-axis.

5. Can the quadratic formula be used for all quadratic equations?

Yes, the quadratic formula can be used to solve any quadratic equation, even those that cannot be easily factored. However, it is important to note that the formula only gives the solutions for the equation, not the process of solving it. Other methods may be more efficient or appropriate for certain equations.

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