4x^2/[(1.5-x)(2.0-2x)^2] =
= 2.72 x 10^11
solve for x
To solve the equation for x, we need to isolate the variable on one side of the equation. Here's how you can solve it step by step:
1. Multiply both sides of the equation by the denominator [(1.5-x)(2.0-2x)^2] to remove it from the denominator on the left-hand side. The equation becomes:
4x^2 = (1.5-x)(2.0-2x)^2 * 2.72 x 10^11
2. Expand the right-hand side of the equation using the distributive property:
4x^2 = (1.5-x) * (4.0-8x+4x^2) * 2.72 x 10^11
3. Simplify the equation step by step. Start by multiplying the terms within the parentheses:
4x^2 = (1.5-x) * (4.0-8x+4x^2) * 2.72 x 10^11
= (1.5-x) * (4.0*2.72 x 10^11 - 8x*2.72 x 10^11 + 4x^2*2.72 x 10^11)
4. Distribute the multiplication inside the parentheses:
4x^2 = (1.5-x) * (10.88 x 10^11 - 21.76 x 10^11 x + 10.88 x 10^11 x^2)
5. Simplify the equation further:
4x^2 = 10.88 x 10^11 - 21.76 x 10^11 x + 10.88 x 10^11 x^2 - 1.5 * 10.88 x 10^11 + x * 10.88 x 10^11 - x^2 * 10.88 x 10^11
6. Collect like terms by combining the coefficients of the different powers of x:
4x^2 + 21.76 x 10^11 x + x^2 * 10.88 x 10^11 = 10.88 x 10^11 - 1.5 * 10.88 x 10^11
7. Rearrange the equation by moving all the terms to one side:
4x^2 + 21.76 x 10^11 x + x^2 * 10.88 x 10^11 - 10.88 x 10^11 + 1.5 * 10.88 x 10^11 = 0
8. Combine like terms on the left-hand side:
(4 + 21.76 x 10^11 + 10.88 x 10^11) x^2 + (1.5 * 10.88 x 10^11 - 10.88 x 10^11) x = 0
9. Simplify the equation further:
(36.64 x 10^11) x^2 + (1.5 * 10.88 x 10^11 - 10.88 x 10^11) x = 0
10. Factor out x from the equation:
x ((36.64 x 10^11) x + (1.5 * 10.88 x 10^11 - 10.88 x 10^11)) = 0
11. Apply the zero-product property: For the equation to be true, either x or the polynomial in the parentheses must be equal to zero.
12. Set each factor equal to zero:
x = 0
(36.64 x 10^11) x + (1.5 * 10.88 x 10^11 - 10.88 x 10^11) = 0
13. Solve the second equation for x:
(36.64 x 10^11) x + (1.5 * 10.88 x 10^11 - 10.88 x 10^11) = 0
(36.64 x 10^11) x = - (1.5 * 10.88 x 10^11 - 10.88 x 10^11)
(36.64 x 10^11) x = 10.88 x 10^11 - 1.5 * 10.88 x 10^11
(36.64 x 10^11) x = 9.38 x 10^11
x = (9.38 x 10^11) / (36.64 x 10^11)
x = 0.255 or x ≈ 2.55 x 10^-1
So the solutions for the equation are x = 0 and x ≈ 2.55 x 10^-1.