Bit of help please on this fine Sunday morning
$\displaystyle \frac{x}{3}=\frac{2}{x}$
$\displaystyle \frac{2}{x + 1}=\frac{2x}{2x - 3} + 3$
Rationals never where my strong point
Thank you!
Simplifying
x2 = 6
Solving
x2 = 6
Move all terms containing x to the left, all other terms to the right.
Simplifying
x2 = 6
Take the square root of each side:
x = {2.449489743}
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Simplifying
2(2x + -3) = 2x(x + 1) + 3(x + 1)(2x + -3)
Reorder the terms:
2(-3 + 2x) = 2x(x + 1) + 3(x + 1)(2x + -3)
(-3 * 2 + 2x * 2) = 2x(x + 1) + 3(x + 1)(2x + -3)
(-6 + 4x) = 2x(x + 1) + 3(x + 1)(2x + -3)
Reorder the terms:
-6 + 4x = 2x(1 + x) + 3(x + 1)(2x + -3)
-6 + 4x = (1 * 2x + x * 2x) + 3(x + 1)(2x + -3)
-6 + 4x = (2x + 2x2) + 3(x + 1)(2x + -3)
Reorder the terms:
-6 + 4x = 2x + 2x2 + 3(1 + x)(2x + -3)
Reorder the terms:
-6 + 4x = 2x + 2x2 + 3(1 + x)(-3 + 2x)
Multiply (1 + x) * (-3 + 2x)
-6 + 4x = 2x + 2x2 + 3(1(-3 + 2x) + x(-3 + 2x))
-6 + 4x = 2x + 2x2 + 3((-3 * 1 + 2x * 1) + x(-3 + 2x))
-6 + 4x = 2x + 2x2 + 3((-3 + 2x) + x(-3 + 2x))
-6 + 4x = 2x + 2x2 + 3(-3 + 2x + (-3 * x + 2x * x))
-6 + 4x = 2x + 2x2 + 3(-3 + 2x + (-3x + 2x2))
Combine like terms: 2x + -3x = -1x
-6 + 4x = 2x + 2x2 + 3(-3 + -1x + 2x2)
-6 + 4x = 2x + 2x2 + (-3 * 3 + -1x * 3 + 2x2 * 3)
-6 + 4x = 2x + 2x2 + (-9 + -3x + 6x2)
Reorder the terms:
-6 + 4x = -9 + 2x + -3x + 2x2 + 6x2
Combine like terms: 2x + -3x = -1x
-6 + 4x = -9 + -1x + 2x2 + 6x2
Combine like terms: 2x2 + 6x2 = 8x2
-6 + 4x = -9 + -1x + 8x2
Solving
-6 + 4x = -9 + -1x + 8x2
Reorder the terms:
-6 + 9 + 4x + x + -8x2 = -9 + -1x + 8x2 + 9 + x + -8x2
Combine like terms: -6 + 9 = 3
3 + 4x + x + -8x2 = -9 + -1x + 8x2 + 9 + x + -8x2
Combine like terms: 4x + x = 5x
3 + 5x + -8x2 = -9 + -1x + 8x2 + 9 + x + -8x2
Reorder the terms:
3 + 5x + -8x2 = -9 + 9 + -1x + x + 8x2 + -8x2
Combine like terms: -9 + 9 = 0
3 + 5x + -8x2 = 0 + -1x + x + 8x2 + -8x2
3 + 5x + -8x2 = -1x + x + 8x2 + -8x2
Combine like terms: -1x + x = 0
3 + 5x + -8x2 = 0 + 8x2 + -8x2
3 + 5x + -8x2 = 8x2 + -8x2
Combine like terms: 8x2 + -8x2 = 0
3 + 5x + -8x2 = 0
Factor a trinomial.
(1 + -1x)(3 + 8x) = 0
the factor '(1 + -1x)'
Simplifying
1 + -1x = 0
Solving
1 + -1x = 0
Move all terms containing x to the left, all other terms to the right.
Add '-1' to each side of the equation.
1 + -1 + -1x = 0 + -1
Combine like terms: 1 + -1 = 0
0 + -1x = 0 + -1
-1x = 0 + -1
Combine like terms: 0 + -1 = -1
-1x = -1
Divide each side by '-1'.
x = 1
Simplifying
x = 1
the factor '(3 + 8x)'
Simplifying
3 + 8x = 0
Solving
3 + 8x = 0
Move all terms containing x to the left, all other terms to the right.
Add '-3' to each side of the equation.
3 + -3 + 8x = 0 + -3
Combine like terms: 3 + -3 = 0
0 + 8x = 0 + -3
8x = 0 + -3
Combine like terms: 0 + -3 = -3
8x = -3
Divide each side by '8'.
x = -0.375
Simplifying
x = -0.375
x = {1, -0.375}
It looks like you've correctly smplified the equation to $\displaystyle 8x^2 - 5x - 3= 0$. Your two solutions are correct, although I prefer to use fractions ...... -3/8 rather than -0.375.
It's admirable that you're posting all of your work, every single step. Excellent. But such detail can make it difficult to read and follow the plot ..... Perhaps you could post just the key steps. If more detail is needed someone will ask.