Questions: By completing the square, put the equation 4x^2+y^2+3z^2=22-8x-10y+6z into the standard form a(x-x0)^2+b(y-y0)^2+c(z-z0)^2=d. Then, use your answer to sketch a graph of this quadric surface on paper.

By completing the square, put the equation
4x^2+y^2+3z^2=22-8x-10y+6z
into the standard form a(x-x0)^2+b(y-y0)^2+c(z-z0)^2=d. Then, use your answer to sketch a graph of this quadric surface on paper.

Transcript text: By completing the square, put the equation \[ 4 x^{2}+y^{2}+3 z^{2}=22-8 x-10 y+6 z \] into the standard form $a\left(x-x_{0}\right)^{2}+b\left(y-y_{0}\right)^{2}+c\left(z-z_{0}\right)^{2}=d$. Then, use your answer to sketch a graph of this quadric surface on paper.

Solution

Solution Steps

Step 1: Rearrange the Equation

Rearrange the given equation to group the terms involving the same variable: \[ 4x^2 + 8x + y^2 + 10y + 3z^2 - 6z = 22 \]

Step 2: Complete the Square for $x$

For the $x$ terms: $4x^2 + 8x$

Factor out the 4: \[ 4(x^2 + 2x) \]

Complete the square: \[ 4((x + 1)^2 - 1) = 4(x + 1)^2 - 4 \]

Step 3: Complete the Square for $y$

For the $y$ terms: $y^2 + 10y$

Complete the square: \[ (y + 5)^2 - 25 \]

Step 4: Complete the Square for $z$

For the $z$ terms: $3z^2 - 6z$

Factor out the 3: \[ 3(z^2 - 2z) \]

Complete the square: \[ 3((z - 1)^2 - 1) = 3(z - 1)^2 - 3 \]

Step 5: Substitute Back and Simplify

Substitute the completed squares back into the equation: \[ 4(x + 1)^2 - 4 + (y + 5)^2 - 25 + 3(z - 1)^2 - 3 = 22 \]

Simplify: \[ 4(x + 1)^2 + (y + 5)^2 + 3(z - 1)^2 = 54 \]

Final Answer

The equation in standard form is: \[ 4(x + 1)^2 + (y + 5)^2 + 3(z - 1)^2 = 54 \]

{"axisType": 3, "coordSystem": {"xmin": -10, "xmax": 10, "ymin": -10, "ymax": 10}, "commands": ["4_(x + 1)2 + (y + 5)2 + 3_(z - 1)**2 = 54"], "latex_expressions": ["$4(x + 1)^2 + (y + 5)^2 + 3(z - 1)^2 = 54$"]}

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