Learning about quadratic functions and equations, and I am struggling. It's so easy for others to understand, but math is my weakness, would appreciate all the help you could give. Notes and stuff would be great, as long as I understand.

Question

Quadratic Functions and Equations, Connecting Zeros, Roots, and x-intercepts

Find the roots of the following equations.

a) 2x(x+3) = 0

I got showed this way from a friend, but teacher never taught this way.

My friend divided 2x by 2 and divided 0 by 2, then he got left with x(x+3)=0 and then he got x = 0, x +3 = 0 and got 0, and -3 as answer. But teacher showed us the graphing way and the table of value thing.

Is there an easier way to do this like factoring or something? Want to learn please, and also do we just plug in any number for the table of value thing?

Find Zero of following equation

x^3 + 8x^2 = 20x

Answer 1

Factoring is the easy way, if possible. Not all quadratics can be easily factored, but it is an important method because

If a*b*c*d = 0 then one of a,b,c,d must be zero! That's not true if you set the product to any other number. So, we always set up the equation so that we have a product of factors equal to zero.

In the example you gave, we have

2x(x+3) = 0
so, either
2=0 -- not gonna happen
x=0 -- that's one solution
(x+3)=0 -- so x = -3 is the other solution.

For the other question, we have

x^3 + 8x^2 = 20x

First step: always set things equal to zero

x^3 + 8x^2 - 20x = 0

Now you can see that the x factors out, leaving you with a quadratic:

x(x^2 + 8x - 20) = 0

That's easy to factor, since 10(-2) = -20 and 10 + (-2) = 8

x(x+10)(x-2) = 0

Now we see that either
x=0 -- that's one solution
(x+10)=0 so x = -10 is another
(x-2)=0 so x=2 is the other solution