Mister Exam
(x^2-5x+12)/(x^2-4x+5)>0 inequation
The solution
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2 x - 5*x + 12 ------------- > 0 2 x - 4*x + 5
$$\frac{\left(x^{2} - 5 x\right) + 12}{\left(x^{2} - 4 x\right) + 5} > 0$$
(x^2 - 5*x + 12)/(x^2 - 4*x + 5) > 0
Detail solution
Given the inequality:
$$\frac{\left(x^{2} - 5 x\right) + 12}{\left(x^{2} - 4 x\right) + 5} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x^{2} - 5 x\right) + 12}{\left(x^{2} - 4 x\right) + 5} = 0$$
Solve:
Given the equation:
$$\frac{\left(x^{2} - 5 x\right) + 12}{\left(x^{2} - 4 x\right) + 5} = 0$$
Multiply the equation sides by the denominators:
5 + x^2 - 4*x
we get:
$$\frac{\left(\left(x^{2} - 5 x\right) + 12\right) \left(x^{2} - 4 x + 5\right)}{\left(x^{2} - 4 x\right) + 5} = 0$$
$$x^{2} - 5 x + 12 = 0$$
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
$$a = 1$$
$$b = -5$$
$$c = 12$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{23} i}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{23} i}{2}$$
$$x_{1} = \frac{5}{2} + \frac{\sqrt{23} i}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{23} i}{2}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
$$\frac{\left(0^{2} - 0 \cdot 5\right) + 12}{\left(0^{2} - 0 \cdot 4\right) + 5} > 0$$
so the inequality is always executed
$$\frac{\left(x^{2} - 5 x\right) + 12}{\left(x^{2} - 4 x\right) + 5} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(x^{2} - 5 x\right) + 12}{\left(x^{2} - 4 x\right) + 5} = 0$$
Solve:
Given the equation:
$$\frac{\left(x^{2} - 5 x\right) + 12}{\left(x^{2} - 4 x\right) + 5} = 0$$
Multiply the equation sides by the denominators:
5 + x^2 - 4*x
we get:
$$\frac{\left(\left(x^{2} - 5 x\right) + 12\right) \left(x^{2} - 4 x + 5\right)}{\left(x^{2} - 4 x\right) + 5} = 0$$
$$x^{2} - 5 x + 12 = 0$$
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
$$a = 1$$
$$b = -5$$
$$c = 12$$
, then
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (1) * (12) = -23
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{23} i}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{23} i}{2}$$
$$x_{1} = \frac{5}{2} + \frac{\sqrt{23} i}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{23} i}{2}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
x0 = 0
$$\frac{\left(0^{2} - 0 \cdot 5\right) + 12}{\left(0^{2} - 0 \cdot 4\right) + 5} > 0$$
12/5 > 0
so the inequality is always executed
Solving inequality on a graph