Xv. A Number Is Composed Of Two Digits. The Quotient Is 7 When The Number Is Divided By The Sum Of Its Digits. If 27 Is Subtracted From The Number, The Digits Swap Places. Find The Number. Xvi. If The Roots Of The Quadratic Equation (a - B)x² + (b - C)x + (c - A) = 0 Are Equal, Then Prove That A, B, And C Are In Arithmetic Progression.

Decoding the Two-Digit Number

In the realm of mathematical puzzles, problems involving number properties often present intriguing challenges. This particular problem centers around a two-digit number with specific characteristics. The primary focus is to decipher the digits of this number, leveraging the given clues. At the heart of this puzzle lies the relationship between the number itself, the sum of its digits, and the outcome of certain arithmetic operations. The first piece of information states that when the number is divided by the sum of its digits, the quotient is 7. This gives us a crucial equation that links the number and its constituent digits. The second piece of information introduces another layer of complexity. It tells us that if 27 is subtracted from the number, the digits interchange their places. This provides us with a second equation, allowing us to set up a system of equations that we can solve to find the digits. To effectively tackle this puzzle, we must first represent the two-digit number algebraically. Let's denote the tens digit as 'x' and the units digit as 'y'. Therefore, the number can be expressed as 10x + y. The sum of the digits is simply x + y. Now, we can translate the given information into mathematical equations. The first statement translates to (10x + y) / (x + y) = 7. Multiplying both sides by (x + y) gives us 10x + y = 7x + 7y. Simplifying this equation, we get 3x = 6y, which further reduces to x = 2y. This equation reveals a direct relationship between the tens digit and the units digit. The second statement tells us that when 27 is subtracted from the number, the digits interchange. This can be expressed as 10x + y - 27 = 10y + x. Rearranging the terms, we get 9x - 9y = 27, which simplifies to x - y = 3. Now we have a system of two equations: x = 2y and x - y = 3. We can substitute the first equation into the second equation to solve for y. Substituting x = 2y into x - y = 3, we get 2y - y = 3, which gives us y = 3. Now that we know the units digit, we can find the tens digit using the equation x = 2y. Substituting y = 3, we get x = 2 * 3 = 6. Therefore, the two-digit number is 10 * 6 + 3 = 63. To verify our solution, we can check if it satisfies the given conditions. The sum of the digits is 6 + 3 = 9. Dividing the number by the sum of its digits, we get 63 / 9 = 7, which matches the first condition. Subtracting 27 from the number, we get 63 - 27 = 36, where the digits have indeed interchanged. Thus, our solution is correct. This problem illustrates the power of algebraic representation in solving number puzzles. By translating the given information into equations, we can systematically find the unknown digits. The key is to carefully define variables, express the given conditions as equations, and then solve the resulting system of equations. This approach can be applied to a wide range of similar number problems, making it a valuable problem-solving technique in mathematics. The problem also highlights the importance of verification. Once we have a solution, it's crucial to check if it satisfies all the given conditions. This helps to ensure that our solution is correct and that we haven't made any errors in our calculations. In conclusion, the solution to this two-digit number puzzle is 63. This problem demonstrates the elegance and power of mathematics in solving seemingly complex puzzles, and emphasizes the importance of careful problem-solving techniques and verification.

Exploring Roots of Quadratic Equations

Moving on to another fascinating area of mathematics, let's delve into quadratic equations and their roots. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The roots of a quadratic equation are the values of the variable (typically denoted as 'x') that satisfy the equation. In other words, they are the points where the graph of the quadratic equation intersects the x-axis. Understanding the nature and properties of these roots is a fundamental concept in algebra. The problem at hand involves a specific quadratic equation, (a - b)x² + (b - c)x + (c - a) = 0, and asks us to determine the nature of its roots. The key to solving this problem lies in understanding the discriminant of a quadratic equation. The discriminant, denoted as Δ (Delta), is given by the formula Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation. The discriminant provides valuable information about the nature of the roots. If Δ > 0, the equation has two distinct real roots. This means that the graph of the quadratic equation intersects the x-axis at two different points. If Δ = 0, the equation has exactly one real root (or two equal real roots). This means that the graph of the quadratic equation touches the x-axis at one point. If Δ < 0, the equation has no real roots, but it has two complex roots. This means that the graph of the quadratic equation does not intersect the x-axis. Now, let's apply this knowledge to the given equation, (a - b)x² + (b - c)x + (c - a) = 0. Here, the coefficients are A = (a - b), B = (b - c), and C = (c - a). To find the discriminant, we substitute these values into the formula Δ = B² - 4AC. Δ = (b - c)² - 4(a - b)(c - a) Expanding this expression, we get: Δ = b² - 2bc + c² - 4(ac - a² - bc + ab) Δ = b² - 2bc + c² - 4ac + 4a² + 4bc - 4ab Δ = 4a² + b² + c² - 4ab + 2bc - 4ac At this point, we might be tempted to try to simplify this expression further. However, a clever observation can significantly simplify our task. Notice that if we substitute x = 1 into the original equation, we get: (a - b)(1)² + (b - c)(1) + (c - a) = a - b + b - c + c - a = 0 This means that x = 1 is a root of the equation. This is a crucial piece of information. Since x = 1 is a root, we know that (x - 1) is a factor of the quadratic equation. Let the other root be denoted as x₂. 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. Therefore, 1 + x₂ = -(b - c) / (a - b) x₂ = - (b - c) / (a - b) - 1 x₂ = (-b + c - a + b) / (a - b) x₂ = (c - a) / (a - b) And 1 * x₂ = (c - a) / (a - b) x₂ = (c - a) / (a - b) This confirms our previous result for x₂. Since we have found two real roots (1 and (c - a) / (a - b)), we can conclude that the discriminant must be greater than or equal to zero. In fact, since we have found two distinct roots (unless c - a = a - b), the discriminant is strictly greater than zero. Therefore, the quadratic equation (a - b)x² + (b - c)x + (c - a) = 0 has two real roots. This problem illustrates the importance of understanding the properties of quadratic equations and their roots. The discriminant is a powerful tool for determining the nature of the roots, and clever observations can often simplify the solution process. The recognition that x = 1 is a root of the equation significantly simplified our task, allowing us to find the other root and conclude that the equation has two real roots. In summary, exploring the roots of quadratic equations is a fundamental aspect of algebra. Understanding the discriminant and other properties of quadratic equations allows us to solve a wide range of problems and gain deeper insights into the behavior of these equations.