Introduction to Quadratic Equations and Interrelated Roots

Understanding the Problem

The statement "one less than its value" is indeed confusing. If we interpret the problem correctly, we should consider the roots of the quadratic equation x2 - px q 0. Specifically, we need to find the possible values of the coefficients p and q under the condition that one of the roots is one less than the other. Let's explore this more precisely.

Formulating the Quadratic Equation

Let's denote the roots of the equation x2 - px q 0 as a and a-1. This means the roots are a and a-1. We can use the sum and product of the roots of a quadratic equation to find the coefficients p and q.

Sum and Product of the Roots

The sum of the roots of a quadratic equation x2 - px q 0 is given by a (a-1) 2a - 1. According to Vieta's formulas, the sum of the roots is also equal to p. Therefore:

[math]p a (a-1) 2a - 1[/math]

The product of the roots is given by a * (a-1) a^2 - a. According to Vieta's formulas, the product of the roots is also equal to q. Therefore:

[math]q a^2 - a[/math]

Converting to a Standard Quadratic Form

Now, we know that the quadratic equation can be written as:

x2 - (2a-1)x (a2 - a) 0

This is equivalent to the original form x2 - px q 0, where:

[math]p 2a - 1[/math]

[math]q a^2 - a[/math]

Let's solve for specific values of a to find corresponding values of p and q.

Example Calculations

1. If a 2, then:

[math]p 2(2) - 1 3[/math]

[math]q 2^2 - 2 2[/math]

So, one possible set of values is p 3 and q 2.

2. If a 3, then:

[math]p 2(3) - 1 5[/math]

[math]q 3^2 - 3 6[/math]

So, another possible set of values is p 5 and q 6.

3. If a 0, then:

[math]p 2(0) - 1 -1[/math]

[math]q 0^2 - 0 0[/math]

So, another possible set of values is p -1 and q 0.

Therefore, the possible values of p and q can be expressed as:

Formulas for p and q:

[math]p 2a - 1[/math]

[math]q a^2 - a[/math]

where a can be any real number.

Conclusion

The key takeaway is that, given the roots of a quadratic equation being interrelated, we can use the sum and product of the roots to derive the values of the coefficients p and q. By setting a to different values, we can generate a range of possible values for p and q.

Keywords: quadratic equations, interrelated roots, algebraic equations