Quadratic Equations in Vertex Form

[alex2256]

Homework Statement

Find the roots and the vertex for 3x^2 + 5x - 2

Homework Equations

ax^2 + bx + c
a(x - h) + k

The Attempt at a Solution

OK, this is what I attempted.

3x^2 + 5x - 2 = 0
3x^2 + 6x - x - 2 = 0
3x(x+2) - 1(x+2) = 0
x = 1/3 and x = -2

OK, so I found the roots, however, I am having difficulty understanding finding the vertex of THIS problem. The book says the vertex should be (-5/6, -4 1/12), so here's what I did.

3x^2 + 5x - 2 = 0
x^2 + 5/3x - 2/3 = 0
x^2 + 5/3x - (25/36) - (2/3 - 25/35) = 0
(x + 5/6)^2 - 49/36

therefore the vertex is -5/6 and -49/36; obviously I didn't get -4 1/12.

Can somebody help me and explain to me what's going on?

 

[Office_Shredder]

Yup, it's pretty simple. You went from

x²+5/3x-25/36 (emphasis mine EDIT: Actually, a bold minus sign looks no different than a regular one...)

to

(x+5/6)²

if yo factor this out, you'll see you get a +25/36. So your constant term is off because you put the negative 25/36 into the complete the square portion instead of the positive part

 

[Architeuthis Dux]

Firstly, it is important to note that the vertex form of a quadratic equation is given by a(x-h)^2 + k, where (h,k) is the vertex. In your attempt at a solution, you have correctly found the roots of the equation, but your method for finding the vertex is incorrect.

To find the vertex, we can use the formula h = -b/2a, where a and b are the coefficients of x^2 and x respectively. In this case, a = 3 and b = 5, so h = -5/6. Now, to find k, we can substitute the value of h into the equation and solve for k.

3(-5/6)^2 + 5(-5/6) - 2 = k
25/12 - 25/6 - 2 = k
-49/12 = k

Therefore, the vertex is (-5/6, -49/12), which can also be written as -4 1/12.

In your attempt, you have correctly completed the square, but you have made a mistake in the step where you subtracted 2/3 - 25/36. The correct expression should be (2/3)^2 - 25/36, which simplifies to -49/36. This is why you ended up with -49/36 instead of -4 1/12 as the vertex.

I hope this helps clarify the process of finding the vertex of a quadratic equation in vertex form. Remember to always check your calculations and to use the correct formula for finding the vertex.