algebra questions

Question

Question 1
What is the vertex of f(x) = x2 + 3x + 2?

 

Question 2
What is the exact value of sin (22.5 degrees)?

 

Question 3
Solve the system: 2x + 3y = 7, 3x - 5y = 1

Answer 1

1> If the equation is Ax^2 + Bx + C, the vertex occurs where x = -B/2A

 

2> Use the half-angle formula: sin(x/2) = ((1-cos x)/2)^.5 with x=45 degrees

 

3> Multiply the first equation by 5 and the second one by 3.

Add the equations to add out the y.

Then you will find x=2.

Plug x=2 back into the first equation to find y=1.

Answer 2

1) You can complete the square or use -b/2a to find the vertex, which is (h,k)

    y=x²+3x+2, where a=1, b=3, and c=2

    -b/2a=-3/2

     h=-3/2

     replace the x's with -3/2 to get y, which is k

    y=(-3/2)²+3(-3/2)+2

    y=(9/4)+(-9/2)+2

    y=(9/4)+(-18/4)+2

    y=(-9/4)+2

    y=(-2 1/4)+2

    y=(-1/4), which is k

    the vertex is (-3/2, -1/4) or (-1.5,-0.25)

 

2) exact value of sin(22.5º)

    use sin(x/2)=√[(1-cosx)/2] where x=45º because 45/2=22.5

    sin(45/2)=√[(1-cos45º)/2]

                 =√[(1-[√2/2]/2]

                 =√[(2-√2)/2]/2

                 =√([(2-√2)]/4)

                 =√(2-√2)/√4

   sin(22.5º)=√(2-√2)/2

 

3)  2x+3y=7

     3x-5y=1

multiply top equation by 3 and multiply bottom equation by 2 (all terms)

    6x+9y=21

    6x-10y=2

subtract

         19y=19

             y=1

2x+3y=7

2x+3=7

2x=7-3

2x=4

x=2

(2,1) is the solution