Please simplify the expression and solve the equation.Thank You.
v^2+6v+8,/v^2+v-12, devided by v+2/2v-6
If your expression mean:
[ ( v ^ 2 + 6 v + 8 ) / ( v ^ 2 + v -12 ) ] / [ ( v + 2 ) / ( 2 v - 6 ) ]
then:
[ ( v ^ 2 + 6 v + 8 ) / ( v ^ 2 + v -12 ) ] / [ ( v + 2 ) / ( 2 v - 6 ) ] =
( v ^ 2 + 6 v + 8 ) * ( 2 v - 6 ) / [ ( v ^ 2 + v -12 ) * ( v + 2 ) ] =
( 2 v * v ^ 2 + 2 v * 6 v + 2 v * 8 - 6 * v ^ 2 - 6 * 6 v - 6 * 8 ) / ( v * v ^ 2 + v * v -12 * v + 2 * v ^ 2 + 2 * v - 2 * 12 ) =
( 2 v ^ 3 + 12 v ^ 2 + 16 v - 6 * v ^ 2 - 36 v - 48 ) / ( v ^ 3 + v ^ 2 -12 v + 2 v ^ 2 + 2 v - 24 ) =
( 2 v ^ 3 + 6 v ^ 2 - 20 v - 48 ) / ( v ^ 3 + 3 v ^ 2 -10 v - 24 )
Now do long division of polynomials.
Result is 2
So:
[ ( v ^ 2 + 6 v + 8 ) / ( v ^ 2 + v -12 ) ] / [ ( v + 2 ) / ( 2 v - 6 ) ] = 2
I hope you mean:
(v^2+6v+8)/(v^2+v-12) ÷ ((v+2)/(2v-6) )
= (v+2)(v+4)/((v+4)(v-3) * (2(v-3)/(v+2)
= 2(v+2)(v+4)(v-3) / ( (v+4)(v-3)(v+2) )
= 2 , where v ≠ -4,-2, 3
To simplify the expression and solve the equation, we will follow these steps:
Step 1: Simplify each rational expression separately.
Expression 1: v^2 + 6v + 8 / v^2 + v - 12
Factor the numerator and denominator:
(v + 4)(v + 2) / (v - 3)(v + 4)
Cancel out common factors:
(v + 2) / (v - 3)
Expression 2: (v + 2) / (2v - 6)
Simplify the denominator: 2v - 6 = 2(v - 3)
(v + 2) / 2(v - 3)
Step 2: Combine the simplified expressions using division.
((v + 2) / (v - 3)) / ((v + 2) / 2(v - 3))
Invert the second fraction and multiply:
((v + 2) / (v - 3)) * (2(v - 3) / (v + 2))
Cancel out common factors:
The (v + 2) terms cancel out, and we are left with:
2 / 1
Step 3: Simplify the final expression.
The simplified expression is simply 2.
Therefore, the solution to the equation is v = 2.