If 4x^2 = 15^2 − 9^2 for a real number x, what is the positive value of x obtained from this equation?

Introduction / Context:This question tests the ability to use the difference of squares identity and then solve a simple quadratic equation. Instead of computing 15^2 and 9^2 separately in a slow way, you can exploit the identity a^2 − b^2 = (a − b)(a + b). After simplifying the right-hand side, you solve for x by isolating x^2 and taking the square root, remembering that the quadratic equation has two real roots, one positive and one negative.

Given Data / Assumptions:

• The equation is 4x^2 = 15^2 − 9^2.
• x is a real number.
• The options list positive values, so we focus on the positive root.
• All arithmetic is exact, involving only integers and perfect squares.

Concept / Approach:The right-hand side is a difference of squares. Using a^2 − b^2 = (a − b)(a + b), we can write 15^2 − 9^2 = (15 − 9)(15 + 9). This avoids separate squaring and makes the arithmetic straightforward. Once the right-hand side is simplified, we divide both sides by 4 to obtain x^2. Taking square roots then gives x. Because the question gives only positive choices, we choose the positive root that satisfies the equation.

Step-by-Step Solution:Use the identity a^2 − b^2 = (a − b)(a + b) with a = 15 and b = 9.Compute 15^2 − 9^2 = (15 − 9)(15 + 9) = 6 * 24 = 144.So the equation becomes 4x^2 = 144.Divide both sides by 4 to isolate x^2: x^2 = 144 / 4 = 36.Take square roots: x = ±√36 = ±6.Since the options list positive values, we select x = 6.

Verification / Alternative check:Substitute x = 6 into the original equation to confirm. Left-hand side: 4x^2 = 4 * 6^2 = 4 * 36 = 144. Right-hand side: 15^2 − 9^2 = 225 − 81 = 144. Both sides are equal, so x = 6 is correct. The negative root x = −6 also satisfies the equation, but it does not appear among the given answer choices, and the question typically expects the positive root in such contexts.

Why Other Options Are Wrong:

• x = 9 or x = 12 would give x^2 values of 81 or 144, leading to 4x^2 values that are too large compared with 144.
• x = 3 gives x^2 = 9 and 4x^2 = 36, which is too small.
• Only x = 6 yields 4x^2 = 144, matching the simplified right-hand side.

Common Pitfalls:

• Forgetting the identity a^2 − b^2 = (a − b)(a + b) and instead doing longer arithmetic.
• Dividing incorrectly when solving 4x^2 = 144, for example writing x^2 = 144 * 4.
• Ignoring the fact that both ±6 satisfy the equation and misinterpreting the question's expected root.

Final Answer:6