Please show me how to simplify this expression and solve the equation:
(v^2 + 6v+8)/(v^2 + v - 12) / (v+2)/(2v-6)
Thank you very much.
What was posted does not represent an equation. Please check before posting.
If I assume the second / is an equal sign, then the equation reads:
(v^2 + 6v+8)/(v^2 + v - 12) = (v+2)/(2v-6)
To solve this equation, we start by factoring the quadratic expression into:
(v²+6v+8)=(v+2)(v+4)
and
(v²+v-12)=(v+4)(v-3)
also
(2v-6)=2(v-3)
Putting this altogether, equation becomes:
(v+2)(v+4)/(v+4)(v-3)=(1/2)(v+2)/(v-3)
If (v+2)≠0, (v+4)≠0, (v-3)≠0, then we can cancel common factors on left and right sides, we end up with
1=1/2
which means that our assumption is wrong.
(v+4)≠0 because else the left-hand-side will be indeterminate.
(v-3)≠0 because else both sides will be infinite.
The last possible option is (v+2)=0, which makes both sides equal (0=0).
By elimination, the only possible answer is v+2=0, or v=-2.
The way it was typed and following the order of operation, we have:
(v^2 + 6v+8)/(v^2 + v - 12) / (v+2)/(2v-6)
= (v+2)(v+4)/( (v+4)(v-3)(v+2)(2)(v-3) )
= 1/(2(v-3)^2)
If you meant:
(v^2 + 6v+8)/[ (v^2 + v - 12) / ((v+2)/(2v-6)) ]
then we would have
(v+2)(v+4)/[ (v+4)(v-3)/((v+2)/(2)(v-3)) ]
= (v+2)(v+4) * (2(v+2)(v-3)/( (v+4)(v-3) )
= 2(v+2)^2
e.g. 24/3/4
= (24/3) / 4 = 8/4 = 2 , my first interpretation
matching your typed question
http://www.wolframalpha.com/input/?i=calculate+24%C3%B73%C3%B74
or
= 24 / (3/4)
= 24 * (4/3) = 32 , the second interpretation
http://www.wolframalpha.com/input/?i=calculate+24%C3%B7(3%C3%B74)
As to "solving the equation", since you did not have an equation, there is nothing to solve
To simplify the given expression and solve the equation, follow these steps:
Step 1: Factor the numerator and denominator of the first fraction.
The numerator of the first fraction (v^2 + 6v + 8) can be factored as (v + 4)(v + 2).
The denominator of the first fraction (v^2 + v - 12) can be factored as (v - 3)(v + 4).
Therefore, the first fraction becomes [(v + 4)(v + 2)] / [(v - 3)(v + 4)].
Step 2: Factor the second fraction.
The second fraction (v + 2) can't be factored further.
The denominator of the second fraction (2v - 6) can be factored as 2(v - 3).
So, the second fraction becomes (v + 2) / [2(v - 3)].
Step 3: Divide the first fraction by the second fraction.
To divide two fractions, multiply the first fraction by the reciprocal of the second fraction.
Therefore, the expression becomes [(v + 4)(v + 2)] / [(v - 3)(v + 4)] * [2(v - 3)] / (v + 2).
Step 4: Cancel any common factors and simplify expression.
The common factors (v + 4) and (v - 3) cancel out.
So, the expression simplifies to [(v + 2) * 2] / 1, which becomes 2(v + 2).
Step 5: Solve the equation by setting the expression equal to zero.
2(v + 2) = 0.
Step 6: Solve for v.
Divide by 2: (v + 2) = 0.
Subtract 2 from both sides: v = -2.
Therefore, the simplified expression is 2(v + 2), and the solution to the equation is v = -2.
To simplify and solve the given expression, follow these steps:
Step 1: Simplify the expression by multiplying by the reciprocal of the second fraction.
The given expression is:
(v^2 + 6v + 8)/(v^2 + v - 12) / (v + 2)/(2v - 6)
To simplify, multiply the first fraction by the reciprocal of the second fraction:
(v^2 + 6v + 8)/(v^2 + v - 12) * (2v - 6)/(v + 2)
Step 2: Factorize the numerator and the denominator.
Numerator:
v^2 + 6v + 8 = (v + 4)(v + 2)
Denominator:
v^2 + v - 12 = (v + 4)(v - 3)
(v + 4)(v + 2) / (v + 4)(v - 3) * (2v - 6) / (v + 2)
Step 3: Cancel out the common factors.
(v + 4) cancels out in both the numerator and the denominator.
(v + 2) cancels out in both the numerator and the denominator.
The expression simplifies to:
= (2v - 6) / (v - 3) or 2(v - 3) / (v - 3)
Step 4: Further simplify the expression if needed.
The simplified expression is:
= 2, as (v - 3) cancels out in the numerator and denominator.
Therefore, the solution to the simplified equation is 2.