x+6/4x^2 + 3/2x^2 = x+4/2x^2
To solve the equation, we need to get rid of the fractions and simplify both sides so that we can isolate the variable x.
Let's start by finding a common denominator for the fractions. In this case, the common denominator is 2x^2.
On the left side of the equation:
(x + 6) / (4x^2) + (3 / 2x^2)
To make the denominators the same, we need to multiply the first fraction by (2 / 2) and the second fraction by (2x^2 / 2x^2).
(x + 6) / (4x^2) + (3 / 2x^2)
= (2(x + 6)) / (2 * 4x^2) + (3 * 2x^2) / (2x^2 * 2x^2)
= (2x + 12) / (8x^2) + (6x^2) / (4x^4)
On the right side of the equation:
(x + 4) / (2x^2)
To make this fraction have the same denominator as the left side, we need to multiply it by (4x^2 / 4x^2).
(x + 4) / (2x^2)
= (x + 4) * (4x^2) / (2x^2 * 4x^2)
= (4x^3 + 16x^2) / (8x^4)
Now, the equation becomes:
(2x + 12) / (8x^2) + (6x^2) / (4x^4) = (4x^3 + 16x^2) / (8x^4)
To combine the fractions on both sides, we need a common denominator. In this case, the common denominator is 8x^4. Multiply the first fraction by (x^2 / x^2) and the second fraction by (2 / 2) to get the common denominator:
[(2x + 12) * (x^2)] / (8x^2 * x^2) + [(6x^2) * 2] / (4x^4 * 2)
Simplifying further:
(2x^3 + 12x^2) / (8x^4) + (12x^2) / (8x^4) = (4x^3 + 16x^2) / (8x^4)
Now, we can add the fractions on the left side:
(2x^3 + 12x^2 + 12x^2) / (8x^4) = (4x^3 + 16x^2) / (8x^4)
Combine like terms:
(2x^3 + 24x^2) / (8x^4) = (4x^3 + 16x^2) / (8x^4)
Now, the denominators on both sides are the same. Therefore, we can equate the numerators:
2x^3 + 24x^2 = 4x^3 + 16x^2
Simplify the equation further by moving all the terms to one side:
2x^3 - 4x^3 + 24x^2 - 16x^2 = 0
Combine like terms:
-2x^3 + 8x^2 = 0
Factor out a common term:
-2x^2(x - 4) = 0
Now, we have a product of two factors that equals zero. According to the zero product property, for the product to be equal to zero, at least one of the factors must be zero.
So, we have two possibilities:
-2x^2 = 0 (Factor 1)
x - 4 = 0 (Factor 2)
Solving for x:
For Factor 1:
-2x^2 = 0
Divide both sides by -2: x^2 = 0
Take the square root of both sides: x = 0
For Factor 2:
x - 4 = 0
Add 4 to both sides: x = 4
Therefore, the equation x + (6/4)x^2 + (3/2)x^2 = (x + 4)/(2x^2) has two solutions, x = 0 and x = 4.