How does the discriminant work (quadratics)

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In summary, the discriminant in a quadratic can be used to determine the number of real roots in a quadratic equation. It can also be used to solve for an unknown coefficient in certain applications, such as solving linear ordinary differential equations. This is possible because the discriminant is an explicit expression that can be evaluated with a set of given constants. When dealing with problems involving determining the range of a parameter in the coefficients, the discriminant can be used as an extra constraint to find the final solution.
  • #1
autodidude
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I remember back when I did a lot of quadratic equations, occasionally I had to find an unknown coefficient and if I remember correctly, generally I had to use the discriminant which gave me another quadratic which I could then use to solve for the coefficient.

My question is why does this work? Why can the discriminant be used in this way?
 
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  • #2
Hey autodidude.

The discriminant in a quadratic well tell you whether the quadratic has two real roots (discriminant > 0), one real root (discriminant = 0) or no real roots (two complex roots with discriminant < 0).

The discriminant is a nice way to show the above and in certain applications, the nature of the above affects results that build on the result of solving these including things in calculus like solving linear ordinary differential equations.

Basically you can show by completing the square that the solution can be found in terms of the discriminant, but the discriminant is usually just an explicit expression that can be evaluated with a set of given constants. This is the core of why the quadratic equation works.

I get a feeling though you are not talking about an ordinary quadratic in the "special occasions". Maybe you could outline the sort of problem you are referring to if the above is not the case?
 
  • #3
I think the OP is referring to questions involving determining the range of a parameter in one more more of the coefficients that result in a particular outcome for the number of real roots.

Something for example like: Find the range of parameter "k" such that the quadratic, [itex]x^2 + (2-k)x + 4[/itex], is positive definite.

Is that the type of problem you were thinking of autodidude?
 
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  • #4
Ohh ok.

Well in that case it's just adding an extra constraint which means you find that constraint to find your original problem.

Think of it like a tree where the leaves of the tree are all the conditions that you have and you work all the way back to the root to get a final solution.

It's more or less just a means to an end.
 
  • #5
uart said:
I think the OP is referring to questions involving determining the range of a parameter in one more more of the coefficients that result in a particular outcome for the number of real roots.

Something for example like: Find the range of parameter "k" such that the quadratic, [itex]x^2 + (2-k)x + 4[/itex], is positive definite.

Is that the type of problem you were thinking of autodidude?

Yep, thanks for posting that!

@chiro: Thanks...that makes sense. I think I was expecting some deeper mathematical result lol
 

Related to How does the discriminant work (quadratics)

1. What is the discriminant in quadratic equations?

The discriminant is a value that is used to determine the nature of the roots of a quadratic equation. It is represented by the symbol Δ and is calculated using the formula Δ = b2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.

2. How does the discriminant help solve quadratic equations?

The discriminant helps to determine the number and nature of the roots of a quadratic equation. If the discriminant is positive, the equation will have two distinct real roots. A discriminant of zero indicates that the equation has one real root, and a negative discriminant means that the equation has no real roots, but two complex roots.

3. What does a positive discriminant value indicate?

A positive discriminant value indicates that the quadratic equation has two distinct real roots. This means that the equation will have two solutions.

4. What does a zero discriminant value indicate?

A zero discriminant value indicates that the quadratic equation has one real root. This means that the equation will have one solution.

5. What does a negative discriminant value indicate?

A negative discriminant value indicates that the quadratic equation has no real roots. This means that the equation will have two complex roots, which can be written in the form of a+bi, where a and b are real numbers and i is the imaginary unit.

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