If 12x = 19^2 − 11^2 for a real number x, what is the value of x obtained by simplifying this equation?

Introduction / Context:
This question is similar to earlier difference-of-squares problems. You must simplify the right-hand side 19^2 − 11^2 and then solve the resulting linear equation 12x = constant. Using the identity a^2 − b^2 = (a − b)(a + b) allows you to compute the difference quickly without doing full squaring first.

Given Data / Assumptions:

• The equation is 12x = 19^2 − 11^2.
• x is a real number.
• We are looking for an exact integer value of x.
• All calculations involve simple integers and perfect squares.

Concept / Approach:
Recognise that 19^2 − 11^2 is a difference of squares. Using the formula a^2 − b^2 = (a − b)(a + b), we compute the expression as (19 − 11)(19 + 11). This is much faster than computing 19^2 and 11^2 separately and then subtracting. Once the right-hand side is simplified, solving for x is a one-step division: x = (right-hand side)/12.

Step-by-Step Solution:
Apply the identity with a = 19 and b = 11: 19^2 − 11^2 = (19 − 11)(19 + 11).Compute 19 − 11 = 8 and 19 + 11 = 30.Thus 19^2 − 11^2 = 8 * 30 = 240.The equation becomes 12x = 240.Solve for x by dividing both sides by 12: x = 240 / 12 = 20.

Verification / Alternative check:
We can verify by computing the squares explicitly. 19^2 = 361 and 11^2 = 121, so 361 − 121 = 240. With 12x = 240, dividing by 12 gives x = 20, which matches our earlier calculation. Substituting x = 20 into the original equation confirms that both sides are equal, so the solution is correct.

Why Other Options Are Wrong:

• 17, 13, 11, and 19 would give 12x values of 204, 156, 132, and 228 respectively, which do not match 240.
• The difference-of-squares computation uniquely yields 240, and only x = 20 makes 12x equal to this value.

Common Pitfalls:

• Attempting to square 19 and 11 mentally and making arithmetic mistakes.
• Forgetting or misapplying the identity a^2 − b^2 = (a − b)(a + b).
• Dividing 240 by 12 incorrectly.

Final Answer:
20