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Similar expressions
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1/(x-3)^2-3/(x-3)-4=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
1 3
-------- - ----- - 4 = 0
2 x - 3
(x - 3)
$$\left(\frac{1}{\left(x - 3\right)^{2}} - \frac{3}{x - 3}\right) - 4 = 0$$
Detail solution
Given the equation:
$$\left(\frac{1}{\left(x - 3\right)^{2}} - \frac{3}{x - 3}\right) - 4 = 0$$
Multiply the equation sides by the denominators:
(-3 + x)^2
we get:
$$\left(x - 3\right)^{2} \left(\left(\frac{1}{\left(x - 3\right)^{2}} - \frac{3}{x - 3}\right) - 4\right) = 0$$
$$- 4 x^{2} + 21 x - 26 = 0$$
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
$$a = -4$$
$$b = 21$$
$$c = -26$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 2$$
$$x_{2} = \frac{13}{4}$$
$$\left(\frac{1}{\left(x - 3\right)^{2}} - \frac{3}{x - 3}\right) - 4 = 0$$
Multiply the equation sides by the denominators:
(-3 + x)^2
we get:
$$\left(x - 3\right)^{2} \left(\left(\frac{1}{\left(x - 3\right)^{2}} - \frac{3}{x - 3}\right) - 4\right) = 0$$
$$- 4 x^{2} + 21 x - 26 = 0$$
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
$$a = -4$$
$$b = 21$$
$$c = -26$$
, then
D = b^2 - 4 * a * c =
(21)^2 - 4 * (-4) * (-26) = 25
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 2$$
$$x_{2} = \frac{13}{4}$$
Sum and product of roots
[src]
sum
2 + 13/4
$$2 + \frac{13}{4}$$
=
21/4
$$\frac{21}{4}$$
product
2*13 ---- 4
$$\frac{2 \cdot 13}{4}$$
=
13/2
$$\frac{13}{2}$$
13/2
The graph