Question

1. write the equation in the standard form of a circle.

$x^{2}+y^{2}+6x - 4y - 3 = 0$
for 11 - 12, consider a circle with center $c(5,8)$ that contains the point $p(2,2)$.

1. what is the radius of the circle?
2. what is the equation of the circle in standard form?
3. look at the following quadratic equations and choose the best approach to solving each of them. then explain why you chose that approach. do not solve the equation. use each one from the word bank once.

word bank
a.) square roots
b.) quadratic formula
c.) completing the square
_____ i.) $2x^{2}-3x + 7 = 0$
explain: _____
_____ ii.) $2x^{2}-72 = 0$
explain: _____
_____ iii.) $x^{2}-4x + 8 = 0$

Explanation:

10.

Step1: Group \(x\) and \(y\) terms

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Step2: Complete the square for \(x\) and \(y\) terms

For \(x\): \((x^{2}+6x)\), add \((\frac{6}{2})^{2}=9\)
For \(y\): \((y^{2}-4y)\), add \((\frac{-4}{2})^{2}=4\)

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11.

Step1: Use the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\)

Here, \(x_1 = 5,y_1 = 8,x_2=2,y_2 = 2\)

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12.

Step1: Use the standard form of a circle \((x - h)^{2}+(y - k)^{2}=r^{2}\)

Here, \(h = 5,k = 8,r = 3\sqrt{5}\)

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13.

I.

Step1: Analyze the equation \(2x^{2}-3x + 7=0\)

The equation is of the form \(ax^{2}+bx + c=0\) (\(a = 2,b=-3,c = 7\)). Since it is a general quadratic equation (not easily factorable and not in a form for square - roots or simple completing the square), we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)

II.

Step1: Analyze the equation \(2x^{2}-72=0\)

We can rewrite it as \(x^{2}=36\). This is of the form \(x^{2}=k\) (\(k = 36\)), so we use the square - roots method \(x=\pm\sqrt{k}\)

III.

Step1: Analyze the equation \(x^{2}-4x + 8=0\)

We can rewrite it as \(x^{2}-4x=-8\). Then complete the square: add \((\frac{-4}{2})^{2}=4\) to both sides \((x - 2)^{2}=-4\)

Answer:

1. \((x + 3)^{2}+(y - 2)^{2}=16\)
2. \(3\sqrt{5}\)
3. \((x - 5)^{2}+(y - 8)^{2}=45\)

13.
I. B. Quadratic Formula; It is a general quadratic equation \(ax^{2}+bx + c = 0\) (\(a
eq0\))
II. A. Square Roots; The equation can be written as \(x^{2}=k\) (\(k = 36\))
III. C. Completing the Square; The equation can be rewritten in the form \((x - h)^{2}=k\) after adding a constant to complete the square for the \(x\) - terms.