Question

The equations \(\mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}=0\) and \(\mathrm{q} \mathrm{x}^{2}+\mathrm{rx}+\mathrm{p}=0(\mathrm{r} \neq 0\) ) have a common root, if (a) \(p+q+r=0\) (c) \(p^{3}+q^{3}+r^{3}=p q+q r+r p+p q r\) (d) \(\frac{\mathrm{p}^{3}+\mathrm{q}^{3}+\mathrm{r}^{3}}{\mathrm{pqr}}=3\)

Short answer

Answer: None of the given conditions match the derived condition. There is no correct answer in the given options.

Step-by-step solution

Write down the given quadratic equations

The quadratic equations are: 1) \(px^2 + qx + r = 0\) 2) \(qx^2 + rx + p = 0\) (where \(r \neq 0\)) The equations have a common root.

Write down the sum and product of roots for each equation

For equation (1), let the roots be \(\alpha\) and \(\beta\). Using Vieta's formulas: - \(\alpha + \beta = -\frac{q}{p}\) - \(\alpha \beta = \frac{r}{p}\) For equation (2), let the roots be \(\alpha\) and \(\gamma\). Using Vieta's formulas: - \(\alpha + \gamma = -\frac{r}{q}\) - \(\alpha \gamma = \frac{p}{q}\)

Find the value of \(\beta\gamma\)

Using the product of roots from both equations, we can find the value of \(\beta\gamma\): - \(\alpha\beta\alpha\gamma = \frac{r}{p} \cdot \frac{p}{q}\) - \(\beta\gamma = \frac{r}{q}\)

Find the sum of roots for both equations

Sum of roots in equations (1) and (2) are: - \(\alpha+\beta+\alpha+\gamma = -\frac{q}{p} - \frac{r}{q}\) - \(2\alpha + \beta + \gamma = -\frac{q}{p} - \frac{r}{q}\) Divide both sides by 2: - \(\alpha + \frac{\beta + \gamma}{2} = -\frac{q}{2p}-\frac{r}{2q}\)

Compare the derived condition with the given conditions

Comparing the found equality with the given conditions: (a) \(p+q+r=0\) (b) \(p^3+q^3+r^3=pq+qr+rp+pqr\) (c) \(\frac{p^3+q^3+r^3}{pqr}=3\) None of the given conditions matches the derived condition. Therefore, there is no correct answer in the given options.

Key concepts

Vieta's Formulas

Vieta's formulas are a set of equations that relate the coefficients of a polynomial to sums and products of its roots. In the context of quadratic equations, these formulas provide a straightforward method to express the sum and product of the roots in terms of the coefficients.

For a general quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta's formulas tell us that the sum of the roots, \(\rho_1 + \rho_2\), and the product of the roots, \(\rho_1 \times \rho_2\), are given by:\[\rho_1 + \rho_2 = -\frac{b}{a}\]and\[\rho_1 \times \rho_2 = \frac{c}{a}\]respectively.

These relationships are essential for solving numerous problems related to quadratic equations, including those where quadratic equations share a common root. When two quadratic equations have a common root, Vieta's formulas are used to find relationships between the coefficients of those equations that arise due to the shared root.

Sum of Roots in Quadratic Equations

The sum of the roots in quadratic equations, often represented by \(\rho_1 + \rho_2\), can be calculated directly from the coefficient of the first-degree term (the \(x\) term) in the quadratic equation. According to Vieta's formulas, for the equation \(ax^2 + bx + c = 0\), the sum of the roots is \( -\frac{b}{a} \).

This concept becomes particularly useful when dealing with two quadratic equations that supposedly have a common root. By comparing the sum of roots for each equation, one can potentially find the value of the common root or infer properties about the relations of the coefficients. For example, if two quadratic equations are known to have one root in common, and you've determined the sum of roots for both, you may equate the respective expressions for the root common to both equations to derive further insights or relationships.

Product of Roots in Quadratic Equations

Similarly, the product of the roots in quadratic equations, indicated by \(\rho_1 \times \rho_2\), is given by the ratio of the constant term to the leading coefficient in the quadratic equation. For \(ax^2 + bx + c = 0\), the product of the roots is \( \frac{c}{a} \) as per Vieta's formulas.

When examining the connection between two quadratic equations with a common root, understanding the product of the roots can be revealing. In such scenarios, the product of the roots, apart from the common root, could provide a link between the constant terms and the leading coefficients of the two equations. This connection allows for comparison between the equations which can lead to the discovery of specific identities or to conclude if certain conditions hold true for the coefficients of these equations. The product of roots becomes a vital piece in the puzzle of understanding the intricate relationships that govern the algebraic structure of polynomials.