Michael's Responses in WyzAnt Answers

Hi Amber,

There are two main forms that you can write a line in:

Point-Slope: y-y₁ = m(x-x₁)
We use point-slope when we know the slope m and a point (x₁, y₁)

Slope-Intercept: y = mx + b
We use slope-intercept when we know the slope m and the y-intercept b

There is no fundamental difference between the lines produced by these two equations - in other words, we can convert between Slope-Intercept and Point-Slope form with a little bit of algebra.  

Sometimes, however, the problems don't give us all the bits we need for either form directly, but we have to figure them out.  That is the situation in this case, where we are given two points, but not the slope.

So the first step is to calculate the slope:

m = (y2 - y1) / (x2 - x1)
  = (15 - -2) / (-3 - 14)
  = 17 / -17
  = -1

Now, since we know the slope, and we ALSO know two different points, we can plug one of the points and the slope into the point-slope formula:

y-y1 =  m(x-x1)  Starting equation
y-15 = -1(x- -3) Plug in m=-1 and (x1,y1) = (-3,15)
y-15 = -1(x+3)   Minus a negative -> add a positive

But since the problem wants the answer in slope-intercept form, we must convert to this form using our basic algebra rules:

y-15    = -1(x + 3)    Starting equation
y-15    = -x - 3       Distribute -1 across (x + 3)
y-15+15 = -x - 3 + 15  Add 15 to both sides
y       = -x + 12      Combine terms, DONE

So the line, in slope-intercept form, is y = -x + 12.

Now you might ask why I plugged in (-3, 15) instead of (14, -2).  Well it doesn't actually matter - we could have plugged in either point and we would have gotten the same answer in the end.

The Big Bang theory posits that matter (and anti-matter) were both created as part of the Big Bang.  In other words, matter didn't cause the big bang, matter was produced BY the big bang.  This is a result of the equivalence of matter and energy (Albert Einstein's E=mc²).

However, you also seem to be asking about Darwinism, which has absolutely nothing to do with the Big Bang theory.  Darwinism is an explanation for life on planet Earth, and Big Bang is an explanation for the existence of the universe as a whole - two very fundamentally different questions.  

Daniel posted a good answer....   to complement his answer, I'd also point out that there is a way to factor this even without factoring out the three first.  The key is to realize that you don't have to restrain yourself to whole numbers, but you can use square roots.  Thus, the following is also an acceptable answer:

  (√3 m + √12)(√3 m - √12)
= (√3 m + 2√3)(√3 m - 2√3)

That is "square root of 3, times m, plus two times square root of 3" and "square root of 3, times m, minus two times square root of 3".  Both would yield the same answers for m when put into an equation.

This is an important technique when the problem doesn't simplify to a "perfect square" as yours does after factoring out the three.  For example, a simpler example is:

m2 - 2 = (m + √2)(m - √2)


(To be clear, this is NOT the answer to your question - it is a different example.)

When trying to prove trig identities, it is often helpful to convert TAN functions into SIN/COS functions:

Proof Step 1: Start with the original equation to prove:
tan²x - sin²x = (tan²x)(sin²x)

Proof Step 2: Replace tan with sin/cos
(sin²x/cos²x) - sin²x = (sin²x/cos²x)(sin²x)

Proof Step 3: Obtain a common denominator on left, simplify right
(sin²x - sin²x cos²x) / cos²x = sin⁴x / cos²x

Proof Step 4: Cancel cos²x from both denominators
sin²x - sin²x cos²x = sin⁴x

Proof Step 5: Factor out sin²x from left
(sin²x)(1 - cos²x) = sin⁴x

Proof Step 6: Use trig identity 1-cos²x = sin²x
(sin²x)(sin²x) = sin⁴x

Proof Step 7: Simplify, DONE.
sin⁴x = sin⁴x

Juan,

I believe that you have copied your problem incorrectly.  There are no 4 consecutive EVEN integers that multiply out to 1680.  

You can see this for yourself - lets pick the first 4 even integers: 2, 4, 6, 8.  The product of these even integers is 384 - too small.  The next potential set is 4, 6, 8, 10, and the product of these four is 1920, which is too big. 

Therefore, you must have copied or typed the problem wrong.

Assuming that you didn't mean to say EVEN integers, there is a solution.  If we assign the first number in the sequence to this variable 'n', then the four numbers would be:

    n, n+1, n+2, n+3

The product of these is:

    n(n+1)(n+2)(n+3) = 1680

Which we can FOIL.  I've done this in three steps - first FOILing the first two terms 'n' and 'n+1' and FOILing 'n+2' and 'n+3', then FOILing the two results, and then combining terms and collecting them on the left side of the equation:

    n(n+1)(n+2)(n+3) = 1680
    (n² + n)(n² + 5n + 6) = 1680
    (n⁴ + 5n³ + 6n² + n³ + 5n² + 6n) = 1680
    n⁴ + 6n³ + 11n² + 6n - 1680 = 0

Now we have to solve this equation for n.  There are several ways to do this, none of which are easily described in this forum.  I used a program called "Octave" to find the solutions; it is similar to Matlab.  However, there are two REAL answers: n = -8 and n = 5.

So, ASSUMING that you are NOT looking for EVEN integers, but any integers, the answer is EITHER: (5, 6, 7, 8) OR (-8, -7, -6, -5).  Multiplying either of these sets of numbers results in 1680.  Notice that both sets of numbers are the same, except one is positive and one is negative.  But since it is an even number of integers, the answers to the multiplication are both positive.