Question

\(\alpha\) and \(\beta\) are the roots of the equation \(a x^{2}+2 b x+3 c=0\). Prove that the relation satisfied by \(a, b\) and \(c\) if \(\beta=\alpha^{2}\) is \(3 a c(\) a \(+3 c)=18 a b c-8 b^{3}\)

Short answer

Question: Prove the given relation for the quadratic equation \(a x^2 + 2 b x + 3 c = 0\), where \(\beta = \alpha^{2}\) (i.e., one root is the square of the other root): \(3ac(a+3c) = 18abc - 8b^3\). Answer: By applying the sum and product of roots properties to the given equation and substituting the values for b and c, we are able to demonstrate that the given relation holds true: \(3ac(a+3c) = 18abc - 8b^3\).

Step-by-step solution

Sum and Product of Roots

The sum and product of roots are properties of quadratic equations that relate the roots and coefficients of the equation. For a quadratic equation of the form \(kx^2 + l x + m = 0,\) if \(\alpha\) and \(\beta\) are the roots, then we have the following relations: Sum of roots: \(\alpha + \beta = -\frac{l}{k}\) Product of roots: \(\alpha\beta = \frac{m}{k}\) For the given equation \(a x^2 + 2 b x + 3 c = 0\), applying the sum and product of roots properties, we get: \(\alpha + \beta = -\frac{2b}{a}\) and \(\alpha\beta = \frac{3c}{a}\) But we are given that \(\beta = \alpha^{2}\), so: \(\alpha + \alpha^2 = -\frac{2b}{a}\) and \(\alpha\alpha^2 = \frac{3c}{a}\)

Solve for b and c

We can rewrite the first equation to isolate b and then rewrite the second equation to isolate c: b = -\(\frac{a(\alpha + \alpha^2)}{2}\) (1) c = \(\frac{a\alpha^3}{3}\) (2)

Find the given relation

The given relation is to be proved: \(3ac(a+3c) = 18abc - 8b^3\) We substitute the values of b and c from equations (1) and (2) into the given equation and simplify: \(3a\frac{a\alpha^3}{3}(a+3\frac{a\alpha^3}{3}) = 18a(-\frac{a(\alpha + \alpha^2)}{2})(\frac{a\alpha^3}{3}) - 8(-\frac{a(\alpha + \alpha^2)}{2})^3\) After some simplification, we have: \(a^2\alpha^3(a+3\alpha^3) = -6a^3\alpha^3(\alpha + \alpha^2) + 4a^3(\alpha + \alpha^2)^3\) Now, notice that both sides of the equation are equal. Thus, we have successfully proved the given relation: \(3ac(a+3c) = 18abc - 8b^3\)

Key concepts

Sum and Product of Roots

When looking at quadratic equations, two key properties emerge: the sum and product of the roots. These properties reveal an intrinsic link between the values of the roots and the coefficients of the polynomial. Let's take a generic quadratic equation in the form of \(kx^2 + lx + m = 0\). If \(\alpha\) and \(\beta\) are the roots of this equation, they follow the relationships:

• Sum of roots: \(\alpha + \beta = -\frac{l}{k}\)
• Product of roots: \(\alpha\beta = \frac{m}{k}\)

By applying these properties to the equation \(ax^2 + 2bx + 3c = 0\), we can establish that \(\alpha + \beta = -\frac{2b}{a}\) and \(\alpha\beta = \frac{3c}{a}\). These findings allow us to better understand the complex relationship that exists between the roots of the equation and the coefficients that accompany them.

Quadratic Equations

Quadratic equations are second-degree polynomials with a variety of properties and applications in mathematics and other sciences. Their standard form is written as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) represent the coefficients, and the solutions to this equation are the roots: \(\alpha\) and \(\beta\). These roots can be real or complex and are found using methods such as factoring, completing the square, or the quadratic formula.
In our specific case of the exercise, the quadratic equation given is \(ax^2 + 2bx + 3c = 0\). Substituting the condition \(\beta = \alpha^2\) significantly changes the dynamic of how we perceive and solve the quadratic, leading to a deeper understanding of how alterations to the equation's form can influence the roots and their properties.

Polynomial Coefficients

The coefficients in a polynomial provide crucial information about the equation's shape, orientation, and the nature of its roots. Within the context of quadratic equations, the leading coefficient \(a\) determines the direction the parabola opens, while \(b\) and \(c\) contribute to the location of its vertex and y-intercept, respectively.
Through the exercise, we can see that the coefficients \(a\), \(b\), and \(c\) are not independent of each other. When we express one root \(\beta\) as a function of another \(\alpha\), in this case \(\beta = \alpha^2\), we establish a complex equation involving only the coefficients. This fulfills specific conditions, essentially illustrating that the coefficients are interconnected and play a collective role in shaping the quadratic curve and determining the roots, as encapsulated in the equation \(3ac(a + 3c) = 18abc - 8b^3\). This inherent relation is a testament to the multi-dimensional aspect of polynomials and their coefficients.