Solving Equations: Understanding Factoring and Finding Solutions

[Femme_physics]

So let's say I have an equation

x(x-3)² = 0

I can factor it out to x(x-3)(x-3) = 0

Since there are two 3's in parenthesis, does this mean that the answers are

x₁ = 3
x₂ = 3
and
x₃ = 0

 

[I like Serena]

Yes!

 

[Mark44]

So let's say I have an equation

x(x-3)² = 0

I can factor it out to x(x-3)(x-3) = 0

Since there are two 3's in parenthesis, does this mean that the answers are

x₁ = 3
x₂ = 3
and
x₃ = 0

Yes, but you can also say that the solutions are x = 3 and x = 0. If you want to be picky, you can say that 3 is a root of multiplicity 2 of your equation.

Multiplicity of roots plays a role in the shape of a graph. For y = x(x-3)², the x-intercepts are at x = 0 and x = 3. Since the multiplicity of the root at 0 is one, the graph of this equation looks a lot like the graph of y = x for x near 0.

Since the multiplicity of the root at 3 is two, the graph of this equation looks like the graph of y = (x - 3)² for x near 3. IOW, near 3 the graph of the original equation looks like a parabola whose vertex is at (3, 0), and that opens upward. For values of x that aren't close to 3 and aren't close to 0, the effects of the cubic equation are dominant over the local effects near 0 and 3.

 

[Femme_physics]

Yes!

Wait, so there are 3 solutions, but since two of the solutions are the same, can I say
x₁ = 3
and
x₂ = 0

Without writing 3 two times as the answer?

 

[Femme_physics]

Yes, but you can also say that the solutions are x = 3 and x = 0

That'd be better! And thanks for the rest of the explanation! :)

 

[I like Serena]

Wait, so there are 3 solutions, but since two of the solutions are the same, can I say
x₁ = 3
and
x₂ = 0

Without writing 3 two times as the answer?

Again yes!

You appear to understand perfectly! :)

 

[Femme_physics]

I love it how you always make me feel awesome, ILS, even for simple stuff. You're an exemplary teacher and mentor! :)

 

[I like Serena]

I love it how you always make me feel awesome, ILS, even for simple stuff. You're an exemplary teacher and mentor! :)

I like the feedback
I'm learning here too you know!
Trying to earn a superhero homework helper suit (with a big "HH" at front and undies over costume-- yep) :P