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Featured 3 months ago

Let's turn

Now, we see that

Similarly, we see that

Therefore, we have:

Let's graph the two parabolas. When we do, we get:

Now, the blue parabola applies when

Therefore, we have(focus on the red graph):

Now, to find its inverse, we have to reflect the red graph over the line

Let's try this mathematically.

Our piece-wise function was:

Let's find the inverse function for each situation.

Now, remember that

Therefore, we can now find the inverse functions.

=>

=>

Similarly,

=>

=>

=>

We now graph these two sideways parabolas:

When

When

We therefore have:

Just note that

Featured 2 months ago

**Prerequisite :**

Therefore, in our **Problem,**

**Respected Lorenzo D. has already derived!.**

Featured 2 months ago

Our line is in the form of

Here, we can see

Recall the formula for the slope:

Where

We are given a point through which the line passes,

Since

Let's rewrite our slope equation with all of this information:

We now have an equation with one unknown variable,

Featured 2 months ago

As was stated in Nghi N's answer, the polynomial can not be factored... using real numbers.

However, if you are in a more advanced algebra course, and you have met complex numbers, see below.

**Problem:**

Factor:

The discriminant is negative

**However, the quadratic formula gives roots, which can then be written in the form of factors.**

**If r is a root for a polynomial expression, then (x - r) is a factor. This is true, whether or not the roots are real.**

The value of the expression inside the square root is already known,

**The roots:**

**The two factors are:**

and

**Here's where the complex numbers appear:**

Substitute the value above to obtain the two "complex factors" of the polynomial:

Featured 2 months ago

We have:

Now, for a product to be less than zero, either one factor must be less than zero and the other greater than zero, i.e. either one factor is positive and one is negative, or vice versa:

or

The first case is not possible (

Therefore, the solution to the inequality is

"Negative" and "positive" infinity aren't included in the answer because

Featured 1 month ago

Given:

#x^3+y^3#

Note that if

#x^3+y^3=(x+y)(x^2-xy+y^2)#

We can calculate the discriminant for the remaining homogeneous quadratic in

#x^2-xy+y^2#

is in standard form:

#ax^2+bxy+cy^2#

with

This has discriminant

#Delta = b^2-4ac = color(blue)(1)^2-4(color(blue)(1))(color(blue)(1)) = -3#

Since

We can factor it with complex coefficients by completing the square and using

#x^2-xy+y^2 = (x-1/2y)^2+3/4y^2#

#color(white)(x^2-xy+y^2) = (x-1/2y)^2+(sqrt(3)/2 y)^2#

#color(white)(x^2-xy+y^2) = (x-1/2y)^2-(sqrt(3)/2 y i)^2#

#color(white)(x^2-xy+y^2) = ((x-1/2y)-sqrt(3)/2i y)((x-1/2y)+sqrt(3)/2i y)#

#color(white)(x^2-xy+y^2) = (x-(1/2+sqrt(3)/2i)y)(x-(1/2-sqrt(3)/2i)y)#

So:

#x^3+y^3 = (x+y)(x^2-xy+y^2)#

#color(white)(x^3+y^3) = (x+y)(x-(1/2+sqrt(3)/2i)y)(x-(1/2-sqrt(3)/2i)y)#

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