Question

problem 8
(fill in the blank) : rewrite the following equation for a quadric surface in standard form and name the quadric surface.
$-4x^2 + 25y^2 + z^2 = 100$
standard form of the equation : <your answer here>
name of the quadric surface = <your answer here>

problem 9
(fill in the blank) : complete the squares to rewrite the following equation for a quadric surface in standard form and name the quadric surface.
$4x^2 + y^2 + z^2 - 16x + 4y + 6z - 7 = 0$
for full credit, please show the relevant calculations including the steps to complete the squares.
standard form of the equation : <your answer here>
name of the quadric surface = <your answer here>

Explanation:

(Problem 8):

Step1: Divide by 100 to normalize

$\frac{-4x^2}{100} + \frac{25y^2}{100} + \frac{z^2}{100} = \frac{100}{100}$

Step2: Simplify each fraction

$-\frac{x^2}{25} + \frac{y^2}{4} + \frac{z^2}{100} = 1$

(Problem 9):

Step1: Group like variables

$(4x^2 - 16x) + (y^2 + 4y) + (z^2 + 6z) = 7$

Step2: Factor leading coefficients

$4(x^2 - 4x) + (y^2 + 4y) + (z^2 + 6z) = 7$

Step3: Complete the squares

$4(x^2 -4x +4) + (y^2 +4y +4) + (z^2 +6z +9) = 7 + 16 + 4 + 9$

Step4: Rewrite as perfect squares

$4(x-2)^2 + (y+2)^2 + (z+3)^2 = 36$

Step5: Divide by 36 to normalize

$\frac{(x-2)^2}{9} + \frac{(y+2)^2}{36} + \frac{(z+3)^2}{36} = 1$

Answer:

(Problem 8):
Standard form of the equation: $-\frac{x^2}{25} + \frac{y^2}{4} + \frac{z^2}{100} = 1$
Name of the quadric surface: Hyperboloid of one sheet

---