Please help me simplify the expressions and solve the equations?
1. v^2 + v - 12/ v^2 + 6v + 8 divided by 2v - 6/ v +2
2. 2/ b-2 = b/ b^2 - 3b +2 + b/ 2b - 2
3. 1/ x-2 + 2x/ (x-2)(x-8) = x/ 2(x-8)
4. (m^2 + 2m + 1/ m^3 + 3m^2 + 3m +1 (m^2/m - 3m/3)
Thanks!
Just as in the previous two posts, you need to factor the trinomials and then cancel factors which occur in the top and bottom.
Have you no ideas at all on factoring these?
Or multiplying by the GCD to clear fractions?
For the first one, I can kinda see how I could factor v^2 + v - 12.. Would it look something like 2(v -6)? Which is like the numerator on the other side.
In the last one, I tried just cancelling out factors and got 2m^2/12m^3? Is that close, or no?
well, 2(v-6) = 2v-12
Guess that's not it, eh?
How about (v+4)(v-3)
You need to find two numbers of opposite sign, which multiply to -12 and add up to +1.
On the last one,
(m^2 + 2m + 1/ m^3 + 3m^2 + 3m +1 (m^2/m - 3m/3)
unless you can factor the polynomials, there's nothing to cancel. You can't just toss away terms. What you have is
(m+1)^2 / (m+1)^3 = 1/(m+1)
m^2/m = m
3m/3 = m
m-m = 0
Not sure how those two are stitched together, if they are.
Any luck on the others?
No... I'm confused. Is this going to be useful later in life?
Only you will know that.
It's actually surprising how often basic math skills come in handy.
Other than that, the mental skills developed when solving problems are definitely useful later in life, no matter the field of endeavor.
Now, if you plan to find a sugar daddy/mama and be kept in luxury, maybe not so much... :-)
Sounds like you could profit from some in-person tutoring for a while. Like anything else, math becomes more enjoyable the more familiar you are with it. Think of something you like to do. You probably spend lots of time doing it, because it's fun and easy. That was always math for me. Even when it was not easy, it was fun, and I knew I'd get it one day.
It's not fun for me. It's stressful, and it takes a lot of time. Time I could spend on American literature, History, or learning French. Your sugar mama/daddy joke is funny, but I don't see where these equations and expressions will fit into my life. I'm tired of having stuff crammed down my throat just so that the tests can be passed and then forgotten. I'm tired of teachers saying that the reason we learn this material is because "the state tells us to". You only get so much life and I'm kinda sick of wasting mine.
If Literature, history and French are your life loves, then algebra probably will not be a great part of your life. But, life consists of a wide variety of experiences, and I guarantee you that some basic math know-how (yes, even including algebra) will make things easier. So, tough out the minimum requirements, because they really are the minimum skills you will need to work through life's problems.
No, really.
Stop laughing!!
Sure! I'll help you simplify the expressions and solve the equations step by step.
1. Simplifying the expression (v^2 + v - 12) / (v^2 + 6v + 8) ÷ (2v - 6) / (v + 2):
To divide fractions, we multiply by the reciprocal of the second fraction. So, the expression becomes:
(v^2 + v - 12) / (v^2 + 6v + 8) * (v + 2) / (2v - 6)
To factorize the quadratic expressions in the numerators and denominators:
(v^2 + v - 12) = (v + 4)(v - 3)
(v^2 + 6v + 8) = (v + 2)(v + 4)
(2v - 6) = 2(v - 3)
Substituting these factorized expressions into the simplified expression:
[(v + 4)(v - 3) / (v + 2)(v + 4)] * [(v + 2) / 2(v - 3)]
Now, we can cancel out the common factors:
[(v + 4)(v - 3)] / [(v + 2)(v + 4)] * [(v + 2) / 2(v - 3)]
= (v - 3) / (v + 2) * 1/2
Simplifying further, we have:
(v - 3) / (2v + 4)
= (v - 3) / 2(v + 2)
2. Solving the equation 2 / (b - 2) = b / (b^2 - 3b + 2) + b / (2b - 2):
First, let's simplify the right side expressions by finding the common denominator:
(b^2 - 3b + 2) = (b - 1)(b - 2)
(2b - 2) = 2(b - 1)
The equation now becomes:
2 / (b - 2) = b / [(b - 1)(b - 2)] + b / 2(b - 1)
Multiplying the terms on the right side by their respective denominators:
2 / (b - 2) = (2b + b(b - 2)) / [(b - 1)(b - 2)]
To get rid of the denominators, let's multiply both sides of the equation by (b - 2)(b - 1):
2(b - 2)(b - 1) = 2b + b(b - 2)
Expanding and rearranging the equation:
2(b^2 - 3b + 2) = 2b + b^2 - 2b
Simplifying and solving for b:
2b^2 - 6b + 4 = 2b + b^2 - 2b
b^2 - 6b + 4 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
3. Solving the equation 1 / (x - 2) + 2x / [(x - 2)(x - 8)] = x / [2(x - 8)]:
Multiplying every term by the common denominator (x - 2)(x - 8):
(x - 8) + 2x = x(x - 2)
x - 8 + 2x = x^2 - 2x
Combining like terms and rearranging the equation:
3x - 8 = x^2 - 2x
x^2 - 5x + 8 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
4. Simplifying the expression [m^2 + 2m + 1 / m^3 + 3m^2 + 3m + 1] * [m^2 / m - 3m / 3]:
To simplify, let's factorize the quadratic expression in the numerator and denominator:
(m^2 + 2m + 1) = (m + 1)^2
(m^3 + 3m^2 + 3m + 1) = (m + 1)^3
(m - 3m) = m(1 - 3)
The expression becomes:
[(m + 1)^2 / (m + 1)^3] * [m^2 / (m - 3m) / 3]
We can cancel out the common factors:
[(m + 1)^2 / (m + 1)^3] * [m^2 / -2m / 3]
= (m + 1) / (m + 1)^2 * m / (-2 / 3)
Further simplifying, we have:
(m + 1) / [m(m + 1)] * m / (-2 / 3)
= m / [m(m + 1)] * m / (-2 / 3)
= -3m^2 / 2(m + 1)
These are the steps to simplify the expressions and solve the equations you provided. Make sure to double-check the work and solve the quadratic equations using the appropriate methods.