Question

1. a student solves the equation ( x^2 - 3x = 9x - 12 ). the table shows the student’s work. step equation given ( x^2 - 3x = 9x - 12 ) 1 ( x^2 - 12x = -12 ) 2 ( x^2 - 12x + 36 = -12 + 36 ) 3 ( (x - 6)^2 = 24 ) 4? 5 ( x = 6 - sqrt{24} ) or ( x = 6 + sqrt{24} ) which equivalent equation could complete the table correctly? options (not fully visible but related to solving the quadratic, e.g., ( x^2 - 111 ), ( x^2 = 30 ), ( x - 6 = pmsqrt{30} ), ( x = 6 pmsqrt{24} ) etc.)

Explanation:

Step1: Analyze the Completing the Square Process

The original equation is \(x^{2}-3x = 9x - 12\). First, we combine like terms. Subtract \(9x\) from both sides and add \(12\) to both sides: \(x^{2}-3x-9x + 12=0\), which simplifies to \(x^{2}-12x+12 = 0\) (Step 1: \(x^{2}-12x=- 12\)). Then, for completing the square, we take the coefficient of \(x\) which is \(- 12\), divide by \(2\) to get \(-6\), and square it to get \(36\). So we add and subtract \(36\) or add \(36\) to both sides. Let's check Step 2: \(x^{2}-12x + 30=-12 + 30\). Wait, actually, the correct step for completing the square after \(x^{2}-12x=-12\) is to add \((\frac{-12}{2})^{2}=36\) to both sides. But in the table, Step 3 is \((x - 6)^{2}=24\). Let's expand \((x - 6)^{2}\), we get \(x^{2}-12x + 36\). So from \(x^{2}-12x=-12\), if we add \(36\) to both sides, we have \(x^{2}-12x+36=-12 + 36\), which is \((x - 6)^{2}=24\) (Step 3). Then Step 4 should be solving \((x - 6)^{2}=24\) for \(x\). Taking square roots, \(x - 6=\pm\sqrt{24}\), so \(x=6\pm\sqrt{24}\). Wait, but in the table, Step 5 is \(x = 6-\sqrt{24}\) or \(x=6+\sqrt{24}\), but \(\sqrt{24}\) can be simplified to \(2\sqrt{6}\), but maybe there was a miscalculation earlier. Wait, let's go back to the original equation: \(x^{2}-3x=9x - 12\). Combine like terms: \(x^{2}-12x + 12 = 0\). Using completing the square: \(x^{2}-12x=-12\), add \(36\) to both sides: \(x^{2}-12x + 36=-12 + 36=24\), so \((x - 6)^{2}=24\) (Step 3). Then take square roots: \(x - 6=\pm\sqrt{24}\), so \(x=6\pm\sqrt{24}\). But \(\sqrt{24}=\sqrt{4\times6}=2\sqrt{6}\), but maybe in Step 4, we have \((x - 6)^{2}=24\), so solving for \(x\) gives \(x=6\pm\sqrt{24}\), which is what Step 5 has. Wait, the question is about which equivalent equation could complete the table (Step 4). Let's see the steps:

Step 1: \(x^{2}-12x=-12\)

Step 2: \(x^{2}-12x + 30=-12 + 30\) (this seems like adding 30 to both sides, but that's not completing the square. Wait, maybe a typo, but let's focus on Step 4. After Step 3: \((x - 6)^{2}=24\), Step 4 should be solving for \(x\), so taking square roots: \(x - 6=\pm\sqrt{24}\), so \(x=6\pm\sqrt{24}\). But looking at the options (the ones below the table, maybe? The image shows some options: \(x^{2}=30\), \(x - 111\), \(x=6\pm\sqrt{30}\), \(x=6\pm\sqrt{24}\)). Wait, the correct Step 4 after Step 3 \((x - 6)^{2}=24\) is to take square roots, so \(x - 6=\pm\sqrt{24}\), so \(x=6\pm\sqrt{24}\), which is the last option (the one with \(x=6\pm\sqrt{24}\)). Wait, but maybe I misread. Let's re-express:

Original equation: \(x^{2}-3x=9x - 12\)

Step 1: \(x^{2}-12x=-12\) (subtract \(9x\) and add \(12\) to both sides)

Step 2: \(x^{2}-12x + 30=-12 + 30\) (add 30 to both sides? But that's not standard. Wait, maybe the student made a mistake, but we need to find the equivalent equation for Step 4. After Step 3: \((x - 6)^{2}=24\), Step 4 should be solving for \(x\), so \(x=6\pm\sqrt{24}\), which is the last option (the one with \(x=6\) plus or minus square root of 24). Wait, the options are:

- \(x^{2}=30\)

- \(x - 111\) (maybe \(x^{2}=111\)?)

- \(x=6\pm\sqrt{30}\)

- \(x=6\pm\sqrt{24}\)

So the correct one is \(x=6\pm\sqrt{24}\) because from \((x - 6)^{2}=24\), taking square roots gives \(x - 6=\pm\sqrt{24}\), so \(x=6\pm\sqrt{24}\).

Step2: Verify the Completing the Square

Completing the square for \(x^{2}-12x\): the square of half of -12 is 36, so \(x^{2}-12x + 36=(x - 6)^{2}\). So from \(x^{2}-12x=-12\), add 36 to both sides: \((x - 6)^{2}=24\) (Step 3). Then take square roots: \(x - 6=\pm\sqrt{24}\), so \(x=6\pm\sqrt{24}\) (Step 4, which is the equation to complete the table).

Answer:

\(x = 6\pm\sqrt{24}\) (or the option corresponding to this, likely the last one in the image, e.g., if the options are as I inferred, the correct one is the one with \(x = 6\pm\sqrt{24}\))