Question

SHOW YOUR WORK

A quadratic equation is shown below:

3x2 − 11x + 10= 0

Part A: Find the vertex. Show your work.

Part B: Solve for x using an appropriate method. SHOW THE STEPS OF YOUR WORK.

Answer 1

Answer: x=-16/11

6+11x+10=0

We add all the numbers together, and all the variables

11x+16=0

We move all terms containing x to the left, all other terms to the right

11x=-16

x=-16/11

Answer 2

Vertex:

Roots:
`x=(5)/(3)` or
`x=2`

Explanation:

The given quadratic equation is:

Let
`f(x)=3x^2-11x+10`

We obtain the vertex form by completing the square;

`f(x)=3(x^2-(11)/(3)x)+10`

Add and subtract the square of half the coefficient of x.

`f(x)=3(x^2-(11)/(3)x+(-(11)/(6))^2+10-3(-(11)/(6))^2)`

This simplifies to

`f(x)=3(x-(11)/(6))^2-(1)/(12)`

Hence the vertex is
`((11)/(6),-(1)/(12))`

We now solve to obtain:

`3(x-(11)/(6))^2-(1)/(12)=0`

`3(x-(11)/(6))^2=(1)/(12))`

`(x-(11)/(6))^2=(1)/(36))`

`(x-(11)/(6))=\pm \sqrt{(1)/(36)}`

`x=(11)/(6)\pm (1)/(6)`

`x=(5)/(3)` or
`x=2`