Question

The ratio of the roots of the equation \(\mathrm{x}^{2}-7 \mathrm{x}+1=0\) is the same as the ratio of the roots of the equation \(\lambda \mathrm{x}^{2}+4 \mathrm{x}-5\) \(=0 .\) The value of \(\lambda\) equals (a) \(\frac{16}{245}\) (b) \(\frac{7}{45}\) (c) \(\frac{-245}{16}\) (d) \(\frac{-16}{245}\)

Short answer

a) \(\frac{16}{245}\) b) \(\frac{64}{245}\) c) \(\frac{128}{245}\) d) \(\frac{32}{245}\) Answer: (a) \(\frac{16}{245}\)

Step-by-step solution

Find the ratio of roots for equation 1

Using Vieta's formulas, the sum of the roots of the first equation is \(\frac{7}{1}\), and the product of the roots is \(\frac{1}{1}\) (because the coefficients of the equation \(x^2 - 7x + 1 = 0\) are \(1, -7, 1\)). Let the roots of the first equation be \(\alpha\) and \(\beta\). We know that: 1. \(\alpha + \beta = \frac{7}{1}\) 2. \(\alpha \beta = \frac{1}{1}\) We need to find the ratio \(\frac{\alpha}{\beta}\), which can be done by squaring the first equation, dividing it by the second equation, and rearranging the terms.

Find the ratio of roots for equation 2

Again, we will use Vieta's formulas to find the sum and the product of the roots of the second equation: \(\frac{-4}{\lambda}\) and \(\frac{-5}{\lambda}\) (because the coefficients of the equation \(\lambda x^2 + 4x - 5 = 0\) are \(\lambda, 4, -5\)). We can denote the roots of the second equation as \(\gamma\) and \(\delta\). Therefore, we have: 1. \(\gamma + \delta = \frac{-4}{\lambda}\) 2. \(\gamma \delta = \frac{-5}{\lambda}\) And we need to find the ratio \(\frac{\gamma}{\delta}\).

Equate the two ratios and solve for \(\lambda\)

We are given that \(\frac{\alpha}{\beta} = \frac{\gamma}{\delta}\). Square the first equations for both sets of roots and divide them by the second equations, then equate the ratios and rearrange the terms. \(\frac{(\alpha + \beta)^2}{4\alpha \beta} = \frac{(\gamma + \delta)^2}{4\gamma \delta}\) Substitute the sum and product of roots for each equation. \(\frac{\left(\frac{7}{1}\right)^2}{4\left(\frac{1}{1}\right)} = \frac{\left(\frac{-4}{\lambda}\right)^2}{4\left(\frac{-5}{\lambda}\right)}\) Cross multiply and simplify the equation. \(\lambda\left(\frac{49}{4}\right) = \frac{16}{5} \Rightarrow \lambda = \frac{16}{5} \cdot \frac{4}{49} = \frac{16}{245}\) Thus, the value of \(\lambda\) is \(\frac{16}{245}\). The correct answer is (a) \(\frac{16}{245}\).

Key concepts

Vieta's Formulas

Vieta's formulas are a set of fascinating mathematical relationships. These express the sums and products of the roots of a polynomial equation in terms of its coefficients. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the formulas are quite straightforward:

• The sum of the roots, \( \alpha + \beta \), is given by \( -\frac{b}{a} \).
• The product of the roots, \( \alpha \beta \), is \( \frac{c}{a} \).

These formulas are especially useful for quickly figuring out relationships between roots without having to find their exact numerical values. They can be applied to check whether certain roots have specific characteristics, such as a desired sum or product. When confronted with two equations, like in the given exercise, Vieta's formulas allow one to systematically compare and equate aspects like root ratios by focusing entirely on the coefficients rather than explicitly solving for \( x \). This significantly simplifies the problem-solving process.

Roots of Polynomial Equations

Roots of polynomial equations represent the values for which the polynomial equals zero. For quadratic equations, which take the form \( ax^2 + bx + c = 0 \), there are two roots. These are solutions to the equation that make it true when substituted back into the original expression.
Quadratic equations are quite special because they always have two roots, which could be real or complex, depending on the discriminant \( b^2 - 4ac \). As noted in the previous section, Vieta’s formulas can help us understand these roots better without directly solving for them. Instead, we rely on:

• Sum of the roots: \( \alpha + \beta = -\frac{b}{a} \)
• Product of the roots: \( \alpha \beta = \frac{c}{a} \)

By substituting the expression from one equation into another, we can determine relationships such as ratios of roots or find specific values of coefficients that maintain these root relationships across different polynomial equations.

Ratio of Roots

Determining the ratio of the roots of a polynomial is a central theme in many algebra problems. It involves understanding how the properties of the roots relate compared to one another. This may involve using the relationships from Vieta's formulas to derive the ratio without explicitly solving for each root. In the context of quadratic equations, if we have two roots \( \alpha \) and \( \beta \), the ratio would be expressed as \( \frac{\alpha}{\beta} \).
To find this ratio, we use the relationships derived from the quadratic equation:

• First, square the sum of roots equation and divide it by the product of roots equation.
• Then, reconfigure these expressions to obtain the ratio directly.

In the given exercise, where two different quadratic equations are analyzed, equating the ratios of their respective roots is fundamental to solving for the unknown \( \lambda \). By balancing these ratios with the principles of Vieta's formulas, we are led to the required coefficient, providing a robust solution approach to this type of problems.