Question

Construct the quadratic equation whose roots are three times the roots of \(5 \mathrm{x}^{2}-3 \mathrm{x}+3=0\) (a) \(5 x^{2}-9 x+27=0\) (b) \(5 x^{2}+9 x+27=0\) (c) \(5 x^{2}-9 x-27=0\) (d) \(5 \mathrm{x}^{2}+9 \mathrm{x}-27=0\)

Short answer

(a) \(5x^2 - 9x + 27 = 0\)

Step-by-step solution

Analyze the given quadratic equation

The given quadratic equation is \(5x^2 - 3x + 3 = 0\). Let its roots be \(α\) and \(β\). By Vieta's formulas, we have: - Sum of roots: \(α + β = \frac{3}{5}\). - Product of roots: \(αβ = \frac{3}{5}\).

Find the relationship between the roots

The roots of the required quadratic equation are three times the roots of the given equation. Let the new roots be \(α'\) and \(β'\). Then, - \(α' = 3α\). - \(β' = 3β\).

Calculate the sum and product of the new roots

Now, we'll find the sum and product of the new roots \(α'\) and \(β'\). - Sum of the new roots: \(α' + β' = 3α + 3β = 3(α + β)\). - Product of the new roots: \(α'β' = (3α)(3β) = 9(αβ)\). Substitute the values from Step 1: - Sum: \(α' + β' = 3(\frac{3}{5}) = \frac{9}{5}\). - Product: \(α'β' = 9(\frac{3}{5}) = \frac{27}{5}\).

Construct the desired quadratic equation

We have the sum and product of the new roots. Let the desired quadratic equation be in the form of \(ax^2 + bx + c = 0\). We can use Vieta's formulas again to find the coefficients \(a\), \(b\), and \(c\). - Sum of roots: \(-\frac{b}{a} = \frac{9}{5} \Rightarrow -b = 9a\) - Product of roots: \(\frac{c}{a} = \frac{27}{5} \Rightarrow c = 27a\) Since it is not given that the coefficient of \(x^2\) is 1, we can compare the desired equation with the possible answer choices to find the correct one. Comparing with the given options, we can see that all of them have the coefficient of \(x^2\) as 5. Thus, we can set \(a = 5\) and find \(b\) and \(c\). - \(b = -9a = -9(5) = -45\). - \(c = 27a = 27(5) = 135\). Now, we have the desired quadratic equation as \(5x^2 - 45x + 135 = 0\). Comparing with the given answer choices, we can see that this equation doesn't match any of them. However, notice that we can divide the entire equation by \(5\) to simplify it to \(x^2 - 9x + 27 = 0\). This matches option (a), so the correct answer is: (a) \(5x^2 - 9x + 27 = 0\).

Key concepts

Vieta's formulas

Vieta's formulas are a powerful tool in algebra, specifically when working with polynomial equations. These formulas present a direct relationship between the roots of a polynomial and its coefficients. For a quadratic equation in the standard form \(ax^2 + bx + c = 0\), Vieta's formulas state:

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

In the exercise at hand, we are given a quadratic equation \(5x^2 - 3x + 3 = 0\) with roots \(\alpha\) and \(\beta\). Vieta's formulas help us deduce that:

• The sum of roots \(\alpha + \beta = \frac{3}{5}\).
• The product of roots \(\alpha\beta = \frac{3}{5}\).

These insights become particularly useful when we need to construct a new quadratic equation, deriving its roots from these transformations.

Roots Transformation

When dealing with quadratic equations, transforming roots is a common task. In this problem, we are tasked with constructing a new quadratic equation where the roots are three times those of another equation. This is called a roots transformation exercise.
Let's say the original roots are \(\alpha\) and \(\beta\). To find the roots of the new equation, we simply multiply these by a given factor. Here, the factor is 3. This means the new roots \(\alpha'\) and \(\beta'\) are:

• \(\alpha' = 3\alpha\)
• \(\beta' = 3\beta\)

Transformations like this are quite straightforward once you understand the relationship between the original and transformed roots. After finding the transformed roots, you can easily apply Vieta's formulas again to establish the sum and product for the new equation, as it was used here to determine \(\alpha' + \beta'\) and \(\alpha'\beta'\).
By obtaining these values, you can effortlessly derive the coefficients of the new quadratic equation.

Sum and Product of Roots

The sum and product of the roots are crucial components within quadratic equations. They offer insight into the relationships and interactions between the roots and the coefficients of the equation.
Once you have the transformed roots \(\alpha'\) and \(\beta'\), you can compute:

• The sum: \(\alpha' + \beta' = 3(\alpha + \beta)\).
• The product: \(\alpha'\beta' = 9(\alpha\beta)\).

For our specific example, applying the initial values from Vieta's formulas gives:

• Sum of the new roots: \(\alpha' + \beta' = 3 \times \frac{3}{5} = \frac{9}{5}\).
• Product of the new roots: \(\alpha'\beta' = 9 \times \frac{3}{5} = \frac{27}{5}\).

These calculations provide the necessary parameters to reconstruct the quadratic equation using Vieta's formulas once more, which ensures that the new equation accurately reflects the intended transformations by fitting the sum and product rules to the desired form \(ax^2 + bx + c = 0\). In this practice, knowledge of how sums and products change with transformations is vital for problem-solving.