For Which Value Of Is

For Which Value of Is: Decoding the Mathematical Question

Introduction

The phrase "for which value of is" is a fragment, a common and often frustrating opening to a mathematical problem that appears in textbooks, exams, and online forums worldwide. It is the quintessential call to action in algebra and calculus, signaling that a specific unknown—usually represented by a variable like x, k, or n—must be found to satisfy a given condition. At its core, this incomplete question is an invitation to engage in problem-solving, logical deduction, and algebraic manipulation. It is not a statement but a directive: Determine the specific numerical or algebraic value that makes the following equation, inequality, or condition true. Understanding how to approach and solve these "for which value of" problems is a foundational skill that unlocks higher-level mathematics, physics, engineering, and data science. This article will deconstruct this ubiquitous query, providing a comprehensive roadmap from initial interpretation to final solution, ensuring you can confidently tackle any problem that begins with these four pivotal words.

Detailed Explanation: The Anatomy of the Query

When you encounter the phrase "for which value of [variable] is [condition] true?", you are being asked to solve for the variable that makes a stated mathematical relationship hold. The variable is the unknown quantity we seek. The "condition" is the rule or equation that this unknown must obey. This could be an equation like f(x) = 0 (finding roots/zeros), an inequality like f(x) > 5, a statement about a function's behavior (e.g., "for which value of a is the vertex of the parabola at x=3?"), or a geometric constraint.

The context is almost always algebraic problem-solving. It moves beyond simply "solve for x" in a straightforward equation. Often, the condition involves parameters (constants whose values we need to find) or requires understanding deeper properties of functions, graphs, or sequences. For instance, "For which value of k does the system of equations have no solution?" requires knowledge of linear algebra and the conditions for inconsistency. The phrase forces the solver to think inversely: instead of finding y for a given x, you find the parameter that creates a specific outcome for the relationship between x and y.

Step-by-Step Breakdown: A Systematic Approach

Solving these problems requires a structured methodology. Rushing to guess and check is inefficient and unreliable for complex problems. Follow this logical flow:

1. Identify and Isolate the Target. The first critical step is to clearly identify what you are solving for. The incomplete phrase will be completed in the problem statement: "For which value of k is the expression defined for all real numbers?" Your target is k. Underline it. Recognize it as the parameter or unknown constant you must determine. Distinguish this from other variables (like x) that may be present in the condition. You are not solving for x; you are solving for the value of k that affects the behavior of the expression in x.

2. Translate the Condition into a Mathematical Statement. Precisely convert the English condition into an equation, inequality, or set of conditions. If the problem states, "For which value of a is the line y = 2x + a tangent to the circle x² + y² = 25?", you must translate "tangent" into the geometric condition: the distance from the center of the circle (0,0) to the line equals the radius (5). This yields the equation |a| / √(2² + 1²) = 5. The verbal condition is now a solvable algebraic equation in terms of a.

3. Manipulate Algebraically to Isolate the Target Variable. This is the core computational phase. Using algebraic rules—factoring, expanding, completing the square, applying the quadratic formula, using logarithmic or exponential identities—you must rearrange your mathematical statement so that the target variable (e.g., k, a, c) is isolated on one side. You will treat all other variables (like x) as part of the structure. For example, in "For which value of m does the quadratic equation x² - 4x + m = 0 have exactly one real solution?", you recall that "exactly one real solution" means the discriminant is zero. So, you set b² - 4ac = 0, substitute (-4)² - 4(1)(m) = 0, and solve 16 - 4m = 0 to find m = 4.

4. Interpret the Solution in Context. A solution like k = -2 is meaningless without context. You must verify it satisfies the original condition and state what it means. Does k = -2 make the denominator zero? Does it make the discriminant negative? You must plug it back into the original scenario. For the tangent line example, solving gives a = ±5√5. You must then confirm that for these a values, the system of the line and circle indeed has exactly one solution (a repeated root), confirming tangency.

Real Examples: From Basic to Advanced

Example 1 (Basic Algebra): "For which value of n is the equation 3(x - n) = 2x + 6 true for all values of x?"

• Analysis: "True for all x" means the equation is an identity. The coefficients of x on both sides must be equal, and the constant terms must be equal.
• Process: Expand: 3x - 3n = 2x + 6. Group x terms: (3x - 2x) - 3n = 6 → x - 3n = 6. For this to hold for all x, the coefficient of x must be 1 on the left and 0 on the right? Wait, this reveals a mistake in initial grouping. Better: Bring all terms to one side: 3x - 3n - 2x - 6 = 0 → x - 3n - 6 = 0. For this to be true for every x, the coefficient of x must be 0 (otherwise, different x give different results). So, 1 = 0? That's impossible. Let's re-read: "true for all values of x" means it's an identity. So, 3x - 3n must be identical to 2x + 6. Therefore, coefficients of x: 3 = 2? That's false. There is no value of n that makes 3=2. The problem likely meant "for which value of n is the equation true?" (for some x), or there's a typo. This highlights the importance of precise reading. A corrected version: "For which value of n is x = 2 a solution to 3(x - n) = 2

x + 6?" Now, substitute x = 2: 3(2 - n) = 2(2) + 6 → 6 - 3n = 4 + 6 → 6 - 3n = 10 → -3n = 4 → n = -4/3.

Example 2 (Quadratic Discriminant): "For what value of k does the equation x² + 4x + k = 0 have exactly one real solution?"

• Analysis: "Exactly one real solution" for a quadratic means the discriminant is zero.
• Process: For ax² + bx + c = 0, the discriminant is b² - 4ac. Here, a = 1, b = 4, c = k. Set discriminant to zero: 4² - 4(1)(k) = 0 → 16 - 4k = 0 → 4k = 16 → k = 4.

Example 3 (Advanced - Tangency): "For what value of a is the line y = 2x + 1 tangent to the circle x² + y² = 25?"

• Analysis: A line is tangent to a circle if they intersect at exactly one point. This means the system of equations has exactly one solution.
• Process: Substitute y = 2x + 1 into the circle equation: x² + (2x + 1)² = 25. Expand: x² + 4x² + 4x + 1 = 25 → 5x² + 4x + 1 = 25 → 5x² + 4x - 24 = 0. For exactly one solution, the discriminant of this quadratic in x must be zero: 4² - 4(5)(-24) = 0 → 16 + 480 = 0? That's 496 ≠ 0. This means the line is not tangent for any a in this form. The problem likely intended a different line equation, perhaps y = ax + 1. Let's solve that: Substitute: x² + (ax + 1)² = 25 → x² + a²x² + 2ax + 1 = 25 → (1 + a²)x² + 2ax - 24 = 0. Set discriminant to zero: (2a)² - 4(1 + a²)(-24) = 0 → 4a² + 96(1 + a²) = 0 → 4a² + 96 + 96a² = 0 → 100a² = -96. This has no real solution, indicating an error in the problem setup or a need for a different approach. A correct version might involve a line y = ax + b where b is also unknown, leading to a solvable system.

These examples illustrate that "for what value of" questions demand a systematic approach: understand the condition, translate it into a mathematical constraint, solve for the target variable, and verify the solution in context. Mastery of this process is fundamental to success in algebra and beyond.