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.

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

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

I understand that quadratic functions and equations can be challenging, but I'm here to help you understand them better. Let's break down the questions one by one:

a) 2x(x+3) = 0

Your friend used a method called factoring to solve the equation. Factoring involves finding the common factors and setting each factor equal to zero. In this case, the common factor is x.

By dividing both sides of the equation by 2, your friend simplified it to x(x+3) = 0. Then, by setting each factor equal to zero, x = 0 and x + 3 = 0, they obtained the solutions x = 0 and x = -3.

This is indeed an effective way to solve the equation. Other methods such as graphing and using a table of values can also be used, but factoring is often the simplest and most straightforward approach for quadratic equations.

b) x^3 + 8x^2 = 20x

To find the zeros of this equation, we need to set it equal to zero:

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

Unfortunately, factoring may not be the most efficient method for this equation since it is a cubic equation. However, we can still use factoring by identifying any common factors.

In this case, we can see that x is a common factor, so we can factor it out:

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

Now, we need to solve the quadratic equation x^2 + 8x - 20 = 0. There are a few ways to solve quadratic equations, including factoring, completing the square, and using the quadratic formula.

If factoring doesn't seem straightforward, you can use the quadratic formula to find the roots. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the equation x^2 + 8x - 20 = 0, a = 1, b = 8, and c = -20. Plugging these values into the quadratic formula, we can find the roots of the equation.

Using the quadratic formula, we get:

x = (-8 ± √((8)^2 - 4(1)(-20))) / (2(1))

Simplifying this expression will yield the values of x, which are the roots of the equation.

Using this method, you can find the zeros of the given equation without relying on factoring alone. Remember to always consider different methods and choose the one that suits your comfort level and the complexity of the equation.

I hope this explanation helps you in understanding the process of finding the roots or zeros of quadratic equations better. If you have any further questions, feel free to ask!