Math Problem Statement

Mathematical problems extracted from the uploaded image, which include algebraic equations and inequalities involving parameters.

Solution

The image contains mathematical problems in Polish. Here's a summary of each:

  1. Zadanie 34: Prove that the equation 3x2+(m+2)x+m1=03x^2 + (m + 2)x + m - 1 = 0 has a solution for every value of mRm \in \mathbb{R}.
  2. Zadanie 35: Prove that there is no value of the parameter bb such that the equation x2+(b2)x+b2b+2=0x^2 + (b - 2)x + b^2 - b + 2 = 0 has a solution in the set of real numbers.
  3. Zadanie 36: Determine the number of solutions of the equation (m26)x2(m4)x12=0(m^2 - 6)x^2 - (m - 4)x - \frac{1}{2} = 0 depending on the value of the parameter mm.
  4. Zadanie 37: Find all values of the parameter mm for which the inequality (5m)x22(1m)x+2(1m)<0(5 - m)x^2 - 2(1 - m)x + 2(1 - m) < 0 has no solutions.
  5. Zadanie 38: Find all values of the parameter mm for which the derivative of the function f(x)=1(m2)x2+2(m3)x+5m6f(x) = \frac{1}{\sqrt{(m-2)x^2 + 2(m-3)x + 5m - 6}} is defined in the set of real numbers.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Inequalities
Real Numbers
Function Domains

Formulas

Quadratic equation: ax^2 + bx + c = 0
Discriminant: Δ = b^2 - 4ac
Inequality solving techniques
Function domain rules for square roots

Theorems

The quadratic formula: x = (-b ± √Δ) / 2a
Conditions for real roots of a quadratic equation (Δ ≥ 0)
Domain definition for square root functions

Suitable Grade Level

Grades 10-12