To determine where Tran first made an error in his solution, we need to carefully examine each step in the process of completing the square for the quadratic equation [tex]\(2x^2 - 4x - 3 = 0\)[/tex].
Step 1:
[tex]\[2x^2 - 4x - 3 = 0\][/tex]
Tran adds 3 to both sides of the equation:
[tex]\[2x^2 - 4x = 3\][/tex]
This step is correct.
Step 2:
[tex]\[2(x^2 - 2x) = 3\][/tex]
Tran factors out the 2 from the left side of the equation:
[tex]\[2(x^2 - 2x) = 3\][/tex]
This step is also correct.
Step 3:
In this step, Tran attempts to complete the square. To complete the square inside the parentheses, we need to take half of the coefficient of [tex]\(x\)[/tex] (which is [tex]\(-2\)[/tex]), square it, and then add it inside the parentheses.
Half of [tex]\(-2\)[/tex] is [tex]\(-1\)[/tex], and squaring [tex]\(-1\)[/tex] gives [tex]\(1\)[/tex]:
[tex]\[x^2 - 2x \rightarrow x^2 - 2x + 1\][/tex]
Since we're adding [tex]\(\boxed{1}\)[/tex] inside the parentheses, we must also take it outside the parentheses but keeping in mind that it is inside a term multiplied by 2, so we actually add [tex]\(2 \times 1 = 2\)[/tex] to the right side of the equation:
[tex]\[2(x^2 - 2x + 1) = 3 + 2\][/tex]
In Tran's Step 3, he erroneously added [tex]\(1\)[/tex] instead of [tex]\(2\)[/tex] to the right side of the equation:
[tex]\[2(x^2 - 2x + 1) = 3 + 1\][/tex]
Therefore, the incorrect step is Step 3.
Step 4:
[tex]\[2(x^2 - 2x + 1) = 3 + 2\][/tex]
[tex]\[2(x - 1)^2 = 5\][/tex]
However, due to the error in Step 3, this step would follow through incorrectly. If he had the correct term on the right-hand side, he would write:
[tex]\[2(x-1)^2 = 5\][/tex]
In conclusion, Tran made his first error in Step 3. He should have added [tex]\(2\)[/tex] to the right side, not [tex]\(1\)[/tex].
