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Featured 4 months ago

The first number is 10 and the second number is 7

Let's define the two numbers we are looking for as

The first number added to three times the second number is 31 can be written as:

Three time the first less twice the second is 16 can be written as:

Solve the first equation for

Substitute

Now substitute

Featured 3 months ago

The answer is

Let's rearrange the equation

Let,

Let's do a sign chart

So,

graph{(18-4x)/(x-3) [-32.47, 32.47, -16.23, 16.26]}

Featured 3 months ago

The rule to factorise any quadratic is to find two numbers such that

#"product" = x^2 " coefficient "xx" constant coefficient"#

#"sum" \ \ \ \ \ \ = x " coefficient"#

So for

#"product" = (2)*(-9) = -18#

#"sum" \ \ \ \ \ \ = -3#

So we look at the factors of

# {: ("factor1", "factor2", "sum"), (18,-1,17), (9,-2,7), (6,-3,3), (-18,1,-17),(-9,2,-7), (-6,3,-3) :} #

So the factors we seek are

Therefore we can factorise the quadratic as follows:

# \ \ \ \ \ 2x^2-3x-9 = 2x^2 color(blue)(-6)x + color(green)(3)x -9 #

# :. 2x^2-3x-9 = 2x(x-3) + 3(x -3) #

# :. 2x^2-3x-9 = (2x+3)(x-3) #

This approach works for all quadratics (assuming it does factorise) , The middle step in the last section can usually be skipped with practice.

Featured 3 months ago

Undefined/indeterminate

How to classify

First, as this question is in prealgebra, let's look at it without any reference to higher level mathematics. In this context, we should treat it as undefined.

When we perform the operation *multiplicative inverse* of

As there is no value which can be multiplied by

As the question also mentions division by

In calculus, we consider what happens when values get very large or very close to specific values. Rather than dividing by

#1/1 = 1# #1/(1/2) = 2# #1/(1/10) = 10# #1/(1/10000) = 10000#

#...#

Notice that as the dividend gets closer to

It gets trickier when we have

#0/1 = 0# #0/(1/2) = 0# #0/(1/10) = 0#

#...#

It seems if we just plug in *very close* to certain numbers. Consider another pattern:

#1/1 = 1# #(1/2)/(1/2) = 1# #(1/10)/(1/10) = 1#

#...#

Both the divisor and dividend are getting close to *indeterminate form*.

In general, different branches of math and physics treat

Featured 3 months ago

Note that:

#225x^18y^22 = 15^2*(x^9)^2*(y^11)^2 = (15x^9y^11)^2#

So

We also find:

#(-15x^9y^11)^2 = 225x^18y^22#

So

Note that if

So if, for example,

We can express that we want the non-negative square root by taking the modulus:

#sqrt(225x^18y^22) = abs(15x^9y^11)#

Featured 1 week ago

Vertex

The intercepts are only at 1 point, the origin

The axis of symmetry is the x-axis ie

This is a quadratic in

So instead of form type

Type 1

Type 2

Instead of the horse shoe type shape being up or down it is left or right.

As the coefficient of

Consider equationtype(1): The

Consider equationtype(2); The

However, in this graph

y-intercept is at

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