Question

name:____ teacher:__ hour:____
______ 36. solve ( x^{2}+10 x = 39 ) by completing the square.

a.) ( x=-5 pm sqrt{14} ) b.) ( x = 3 ) and ( x=-13 ) c.) ( x=-3 ) and ( x = 13 ) d.) ( x=5 pm sqrt{14} )
______ 37. write the following function in vertex form and identify the vertex from the following equation ( g(x)=x^{2}+4 x - 7 )
a.) ( (x + 2)^{2}-3 ) vertex: ( (-2,3) ) c.) ( (x + 2)^{2}-3 ) vertex: ( (-2,-3) )
b.) ( (x - 2)^{2}-11 ) vertex: ( (2,-11) ) d.) ( (x + 2)^{2}-11 ) vertex: ( (-2,-11) )
____ 38. what value should be added in the blanks to complete the square? ( x^{2}+8 x+__+y^{2}+10 y + 25=-40+____+25 )
a.) 16 b.) 8 c.) 4 d.) 64
______ 39. write the equation of a circle with center ( c(7,-10) ) and radius ( r = 2 )
a.) ( (x + 10)^{2}+(y - 7)^{2}=4 ) c.) ( (x - 7)^{2}+(y + 10)^{2}=4 )
b.) ( (x - y)^{2}+(7 + 10)^{2}=4 ) d.) ( (x + 7)^{2}+(y - 10)^{2}=4 )
______ 40. what is the center and radius of the circle? ( (x + 5)^{2}+(y - 1)^{2}=36 ).
a.) ( c(5,-1), r = 36 ) b.) ( c(5,-1), r = 6 ) c.) ( c(-5,1), r = 6 ) d.) ( c(-5,1), r = 36 )
______ 41. find the radius of a circle with center ( c(5,6) ) and passes through the point ( p(2,2) )
a.) ( r = 5 ) b.) ( r = 25 ) c.) ( r = 113 ) d.) ( r=sqrt{113} )

Explanation:

Step1: Solve \(x^{2}+10x = 39\) by completing the square

• Add \((\frac{10}{2})^{2}=25\) to both sides: \(x^{2}+10x + 25=39 + 25\)
• Factor the left side: \((x + 5)^{2}=64\)
• Take square roots: \(x+5=\pm8\)
• Solve for \(x\): \(x=-5\pm8\), so \(x = 3\) or \(x=-13\)

Step2: Write \(g(x)=x^{2}+4x - 7\) in vertex form

• Complete the square for \(x^{2}+4x\): add \((\frac{4}{2})^{2}=4\) and subtract 4
• \(g(x)=(x^{2}+4x + 4)-4 - 7=(x + 2)^{2}-11\), vertex is \((-2,-11)\)

Step3: Complete the square for \(x^{2}+8x\)

• Add \((\frac{8}{2})^{2}=16\)

Step4: Write the equation of a circle

• The standard form is \((x - h)^{2}+(y - k)^{2}=r^{2}\), center \((7,-10)\), radius \(2\): \((x - 7)^{2}+(y + 10)^{2}=4\)

Step5: Find center and radius of \((x + 5)^{2}+(y - 1)^{2}=36\)

• Center \((-5,1)\), radius \(\sqrt{36}=6\)

Step6: Find radius of circle with center \((5,6)\) passing through \((2,2)\)

• Use distance formula \(r=\sqrt{(5 - 2)^{2}+(6 - 2)^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\)

Answer:

1. B. \(x = 3\) and \(x=-13\)
2. D. \((x + 2)^{2}-11\), Vertex: \((-2,-11)\)
3. A. \(16\)
4. C. \((x - 7)^{2}+(y + 10)^{2}=4\)
5. C. \(C(-5,1),r = 6\)
6. A. \(r = 5\)