Question

1. error analysis describe and correct the error in writing $y = 4x^2 + 24x - 11$ in vertex form.

$y=4x^2 + 24x - 11$
$y=4(x^2 + 6x) - 11$
$y + 9 = 4(x^2 + 6x + 9) - 11$
$y + 9 = 4(x + 3)^2 - 11$
$y = 4(x + 3)^2 - 20$

Explanation:

Step1: Identify the error

When completing the square inside the parentheses, we factored out a 4 from the first two terms, so when we add 9 inside the parentheses, we are actually adding \(4\times9 = 36\) to the right - hand side. The original solution only added 9 to the left - hand side, which is incorrect.

Step2: Correct the process

Start with \(y = 4x^{2}+24x - 11\)
Factor out 4 from the first two terms: \(y=4(x^{2}+6x)-11\)
To complete the square inside the parentheses, take half of 6 (which is 3), square it (which is 9). But since we have a factor of 4 outside the parentheses, we are adding \(4\times9 = 36\) to the right - hand side. So we add 36 to both sides:
\(y + 36=4(x^{2}+6x + 9)-11\)
Simplify the right - hand side: \(y + 36=4(x + 3)^{2}-11\)
Subtract 36 from both sides to solve for y: \(y=4(x + 3)^{2}-11-36\)
\(y = 4(x + 3)^{2}-47\)

Answer:

The error was in the step of completing the square. When adding 9 inside the parentheses (to complete the square for \(x^{2}+6x\)), since there is a factor of 4 outside the parentheses, we are actually adding \(4\times9 = 36\) to the right - hand side, but the original solution only added 9 to the left - hand side. The correct vertex form is \(y = 4(x + 3)^{2}-47\)