Question

For what value of \(\mathrm{k}\) is one root of the quadratic equation \(9 \mathrm{x}^{2}-18 \mathrm{x}+\mathrm{k}=0\) double the other? (1) 36 (2) 9. (3) 12 (4) 8

Short answer

Question: Find the value of k for which one root is double the other in the quadratic equation \(9x^2 - 18x + k = 0\). Answer: 8

Step-by-step solution

Write down the sum and product of roots

The sum of the roots of a quadratic equation is given by \(-(b/a)\) and the product of the roots is given by \((c/a)\). Using this information: Sum of roots: \(r + 2r = -\frac{-18}{9}\) Product of roots: \(r \cdot 2r = \frac{k}{9}\)

Solve the sum of roots equation

Using the sum of roots equation: \(r + 2r = -\frac{-18}{9}\) \(3r = 2\) \(r = \frac{2}{3}\)

Substitute the value of r in the product of roots equation

Now, let's substitute the value of \(r\) back into the product of roots equation: \(r \cdot 2r = \frac{k}{9}\) \(\frac{2}{3} \cdot 2 \cdot \frac{2}{3} = \frac{k}{9}\) \(\frac{8}{9} = \frac{k}{9}\)

Solve for k

Cross-multiplying and solving for \(k\), we get: \(k = 8\) Thus, the value of \(k\) for which one root is double the other is 8, which corresponds to option (4).

Key concepts

Sum of Roots

Understanding the sum of roots in a quadratic equation is essential for solving problems efficiently. In a standard quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, the roots of the equation can be denoted as \(r_1\) and \(r_2\).

The sum of the roots can be found using the relationship \(r_1 + r_2 = -\frac{b}{a}\). This stems from the symmetry in the coefficients of the quadratic equation when applying the quadratic formula to find the roots. For instance, in the given example, the sum of the roots can be calculated as \(-\frac{-18}{9}\), which simplifies to 2. These relationships help us start towards a solution and analyze the roots’ behavior without needing to find the precise values of the roots first.

Product of Roots

Similar to the sum of the roots, the product of roots is another important characteristic of quadratic equations. The product of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is found by the formula \(r_1 \times r_2 = \frac{c}{a}\).

This implies that no matter what the roots are, their product is always equal to the constant term \(c\), divided by the leading coefficient \(a\). In our exercise, we determine that the product of the roots \(r \cdot 2r = \frac{k}{9}\), which is an essential part of finding the unknown constant \(k\). This product relationship is particularly useful when the roots have special properties or relationships to each other, such as one root being the double of the other.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable \(x\) with a general form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a eq 0\).

Quadratic equations are ubiquitous in mathematics and can describe various phenomena, including projectile motion in physics and profit maximization in economics. They are characterized by their distinct parabolic graphs, which depict the solutions, or 'roots', where the parabola intersects the x-axis. These roots can be real or complex and can have distinctive relationships, such as being mirror images or multiples of each other as presented in our exercise, where we seek a special relationship between the roots.

Solving Quadratic Equations

Solving quadratic equations can be approached through various methods, such as factoring, applying the quadratic formula, completing the square, or graphing. In certain problems like the one presented, where there is a defined relationship between the roots, leveraging the 'sum of roots' and 'product of roots' can lead to a more efficient solution.

By finding the value of one root, which we have denoted as \(r\) in the solution steps, we can use the established relationships to determine the other root and ultimately solve the entire equation. This is a powerful technique in problems where direct factoring is not feasible or where specific numeric values are required for the coefficients.