Question

using the completing-the-square method, find the vertex of the function \\(f(x) = -2x^2 + 12x + 5\\) and indicate whether it is a minimum or a maximum and at what point.

maximum at \\((-3, 5)\\)
minimum at \\((-3, 5)\\)
maximum at \\((3, 23)\\)
minimum at \\((3, 23)\\)

question 2 (multiple choice worth 2 points)
(02.05r mc)

solve \\(x^2 - 8x + 5 = 0\\) using the completing-the-square method.

\\(x = -4 \pm \sqrt{11}\\)
\\(x = 4 \pm \sqrt{11}\\)
\\(x = -4 \pm \sqrt{5}\\)
\\(x = 4 \pm \sqrt{5}\\)

Explanation:

Convert the function to vertex form

Using the Vertex Form of Quadratic knowledge point

$$
LATEXBLOCK0
$$

Determine the vertex and extremum

Using the Vertex Form of Quadratic knowledge point

$$
LATEXBLOCK1
$$

Solve the second equation by completing the square

We isolate the variable terms and complete the square:

$$
LATEXBLOCK2
$$

Answer:

Question 1

• Maximum at \((-3, 5)\)
• Minimum at \((-3, 5)\)
• Maximum at \((3, 23)\) (Correct answer)
• Minimum at \((3, 23)\)

Question 2

• \(x = -4 \pm \sqrt{11}\)
• \(x = 4 \pm \sqrt{11}\) (Correct answer)
• \(x = -4 \pm \sqrt{5}\)
• \(x = 4 \pm \sqrt{5}\)