\(\mathrm{k(k - 18) + 90 = 9}\)What is the solution to the given equation?-{9}36918

1. SIMPLIFY the equation by rearranging terms

• Given: \(\mathrm{k(k - 18) + 90 = 9}\)
• Move all terms to one side: \(\mathrm{k(k - 18) + 90 - 9 = 0}\)
• This gives us: \(\mathrm{k(k - 18) + 81 = 0}\)

2. SIMPLIFY by distributing

• Distribute k through the parentheses: \(\mathrm{k^2 - 18k + 81 = 0}\)
• Now we have a standard quadratic equation

3. INFER the most efficient solution method

• Before using the quadratic formula, check if this factors nicely
• Look at the pattern: \(\mathrm{k^2 - 18k + 81}\)
• Notice: \(\mathrm{81 = 9^2}\), and \(\mathrm{18 = 2(9)}\)
• This suggests a perfect square trinomial: \(\mathrm{(k - 9)^2}\)

4. SIMPLIFY using the perfect square pattern

• Factor: \(\mathrm{(k - 9)^2 = 0}\)
• Take the square root of both sides: \(\mathrm{k - 9 = 0}\)
• Solve: \(\mathrm{k = 9}\)

Answer: D) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when rearranging the original equation.

Many students struggle with: \(\mathrm{k(k - 18) + 90 = 9}\) and incorrectly rearrange to get \(\mathrm{k(k - 18) = 99}\) instead of \(\mathrm{k(k - 18) = -81}\). This leads to \(\mathrm{k^2 - 18k = 99}\), then \(\mathrm{k^2 - 18k - 99 = 0}\), which doesn't factor nicely and produces wrong solutions. When they try to solve this incorrect equation, they may get confused by the messy arithmetic and end up guessing.

Second Most Common Error:

Missing INFER insight about perfect square pattern: Students don't recognize \(\mathrm{k^2 - 18k + 81}\) as a perfect square trinomial.

Instead of seeing the pattern, they immediately jump to the quadratic formula: \(\mathrm{k = \frac{18 \pm \sqrt{324 - 324}}{2}}\). While this works and gives \(\mathrm{k = 9}\), students often make arithmetic errors with the discriminant calculation or misapply the formula, potentially selecting Choice B) 3 or Choice C) 6 from computational mistakes.

The Bottom Line:

This problem rewards students who can systematically manipulate equations and recognize common algebraic patterns. The perfect square trinomial shortcut makes the solution elegant, but careful arithmetic throughout any approach is essential.