Question

Show that two equivalent quadratic forms have the same discriminant.

Short answer

Answer: Yes, two equivalent quadratic forms have the same discriminant.

Step-by-step solution

Recall the definition of the discriminant of a quadratic equation

The discriminant of a quadratic equation of the form ax^2 + bx + c = 0 is given by D = b^2 - 4ac.

Understand the concept of equivalent quadratic forms

Two quadratic forms are equivalent if one can be obtained by a linear transformation of the other. In other words, if we have two quadratic forms f(x) = a_1x^2 + b_1x + c_1 and g(x) = a_2x^2 + b_2x + c_2, these are equivalent if there exists a nonzero constant k such that a_2 = ka_1, b_2 = kb_1, and c_2 = kc_1.

Obtain the discriminant of the two equivalent quadratic forms

Now, let us compute the discriminant for both f(x) and g(x). For f(x): D_1 = b_1^2 - 4a_1c_1 For g(x): D_2 = b_2^2 - 4a_2c_2

Substitute the equivalent quadratic form relationship into the discriminant of g(x)

By definition of equivalent quadratic forms, we have a_2 = ka_1, b_2 = kb_1, and c_2 = kc_1. Now, let's substitute these relationships into the discriminant of g(x): D_2 = (kb_1)^2 - 4(ka_1)(kc_1) = k^2b_1^2 - 4k^2a_1c_1

Compare the discriminants of f(x) and g(x)

Notice that the expression for D_2 can be simplified as follows: D_2 = k^2(b_1^2 - 4a_1c_1) Now, we can see that D_2 is equal to k^2 times the expression for D_1: D_2 = k^2 D_1

Observe the discriminant relationship

As mentioned earlier, k is a nonzero constant. Since k^2 is positive, it does not change the sign of the discriminant. Therefore, we can conclude that both quadratic forms have the same discriminant: Two equivalent quadratic forms have the same discriminant.

Key concepts

Discriminant

The concept of the discriminant is essential in understanding the nature of roots in quadratic equations. For a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is given by the formula \( D = b^2 - 4ac \). This value tells us about the number and type of solutions the quadratic equation has:

• If \( D > 0 \), there are two distinct real roots.
• If \( D = 0 \), there is exactly one real root, indicating a perfect square.
• If \( D < 0 \), there are no real roots, instead resulting in two complex roots.

So, the discriminant doesn't just provide a numerical value, but significant insights into the solution's characteristics. Understanding how the discriminant changes or remains the same under certain conditions, like equivalent quadratic forms, is crucial in deeper mathematical studies.

Equivalent Quadratic Forms

Equivalent quadratic forms come into play when we discuss transformations of quadratic equations. Two quadratic forms, say \( f(x) = a_1x^2 + b_1x + c_1 \) and \( g(x) = a_2x^2 + b_2x + c_2 \), are considered equivalent if one can be transformed into the other using a simple linear transformation. Specifically, this means there exists a non-zero constant \( k \) such that:

• \( a_2 = k a_1 \)
• \( b_2 = k b_1 \)
• \( c_2 = k c_1 \)

This relationship maintains the form of the equation but modifies its coefficients proportionally. Thus, by the relations between these coefficients, equivalent quadratic forms end up having proportional discriminants, revealing that they essentially describe the same mathematical scenario but can be represented differently. This concept is profound because it establishes an equivalence category among quadratic forms based on their fundamental properties.

Linear Transformation

In mathematics, a linear transformation is a mapping between two spaces that preserves the operations of addition and scalar multiplication. When we apply a linear transformation to a quadratic form, it results in a new form that is linearly equivalent to the original. For quadratic forms, linear transformation is performed by adjusting the coefficients of the quadratic expression according to a constant \( k \).Consider again two quadratic forms, \( f(x) = a_1x^2 + b_1x + c_1 \) and \( g(x) = a_2x^2 + b_2x + c_2 \). A linear transformation is given when:

• \( a_2 = k a_1 \)
• \( b_2 = k b_1 \)
• \( c_2 = k c_1 \)

This transformation implies that the structure of the quadratic equation’s graph or solution set doesn't change in essence, though it may appear different due to the change in coefficients. Linear transformations are a powerful tool in mathematics for exploring how different representations relate to each other while preserving the intrinsic properties of objects.