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(x-1)/(2x+3)-(2x-1)/(3-2x)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
x - 1 2*x - 1 ------- - ------- = 0 2*x + 3 3 - 2*x
$$\frac{x - 1}{2 x + 3} - \frac{2 x - 1}{3 - 2 x} = 0$$
Detail solution
Given the equation:
$$\frac{x - 1}{2 x + 3} - \frac{2 x - 1}{3 - 2 x} = 0$$
Multiply the equation sides by the denominators:
3 + 2*x and 3 - 2*x
we get:
$$\left(2 x + 3\right) \left(\frac{x - 1}{2 x + 3} - \frac{2 x - 1}{3 - 2 x}\right) = 0$$
$$\frac{x \left(6 x - 1\right)}{2 x - 3} = 0$$
$$\frac{x \left(6 x - 1\right)}{2 x - 3} \left(3 - 2 x\right) = 0 \left(3 - 2 x\right)$$
$$- 6 x^{2} + x = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -6$$
$$b = 1$$
$$c = 0$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 0$$
$$x_{2} = \frac{1}{6}$$
$$\frac{x - 1}{2 x + 3} - \frac{2 x - 1}{3 - 2 x} = 0$$
Multiply the equation sides by the denominators:
3 + 2*x and 3 - 2*x
we get:
$$\left(2 x + 3\right) \left(\frac{x - 1}{2 x + 3} - \frac{2 x - 1}{3 - 2 x}\right) = 0$$
$$\frac{x \left(6 x - 1\right)}{2 x - 3} = 0$$
$$\frac{x \left(6 x - 1\right)}{2 x - 3} \left(3 - 2 x\right) = 0 \left(3 - 2 x\right)$$
$$- 6 x^{2} + x = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -6$$
$$b = 1$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-6) * (0) = 1
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 0$$
$$x_{2} = \frac{1}{6}$$
Sum and product of roots
[src]
sum
1/6
$$\frac{1}{6}$$
=
1/6
$$\frac{1}{6}$$
product
0 - 6
$$\frac{0}{6}$$
=
0
$$0$$
0