Solve the Inequality for x f(x) = square root of 2x^3-5x^2-3x
The question asks to find the range of values that satisfy the inequality f(x) ≥ 0, where the function f(x) is defined as the square root of a cubic polynomial: 2x^3 - 5x^2 - 3x. The goal would be to determine the set of real numbers x for which the expression under the square root is non-negative (since the square root of a negative number is not real). Solving this inequality involves finding the values of x where the cubic polynomial is either greater than or equal to zero.
$f \left(\right. x \left.\right) = \sqrt{2 x^{3} - 5 x^{2} - 3 x}$
Answer
Solution:
Step 1:
Begin by equating the function to zero: $\sqrt{2x^3 - 5x^2 - 3x} = 0$.
Step 2:
Proceed to isolate $x$.
Step 2.1:
Square both sides to eliminate the square root: $(\sqrt{2x^3 - 5x^2 - 3x})^2 = 0^2$.
Step 2.2:
Carry out the simplification process.
Step 2.2.1:
Express the square root as a power: $(2x^3 - 5x^2 - 3x)^{\frac{1}{2}}$.
Step 2.2.2:
Simplify the left-hand side.
Step 2.2.2.1:
Apply the exponent rules: $(2x^3 - 5x^2 - 3x)^{\frac{1}{2} \cdot 2}$.
Step 2.2.2.2:
Cancel out the common exponent factors: $(2x^3 - 5x^2 - 3x)^{1}$.
Step 2.2.3:
Simplify the right-hand side, noting that $0$ raised to any power is $0$: $2x^3 - 5x^2 - 3x = 0$.
Step 2.3:
Solve the resulting equation for $x$.
Step 2.3.1:
Factor the equation.
Step 2.3.1.1:
Extract $x$ as a common factor: $x(2x^2 - 5x - 3) = 0$.
Step 2.3.2:
Recognize that if any factor equals zero, the equation is satisfied.
Step 2.3.3:
Set $x$ to zero: $x = 0$.
Step 2.3.4:
Solve $2x + 1 = 0$ for $x$.
Step 2.3.4.1:
Subtract $1$ from both sides: $2x = -1$.
Step 2.3.4.2:
Divide by $2$ to solve for $x$: $x = -\frac{1}{2}$.
Step 2.3.5:
Solve $x - 3 = 0$ for $x$: $x = 3$.
Step 2.3.6:
Compile the solutions: $x = 0, -\frac{1}{2}, 3$.
Step 3:
Present the solution in various formats.
Exact Form: $x = 0, -\frac{1}{2}, 3$ Decimal Form: $x = 0, -0.5, 3$
Knowledge Notes:
The problem involves solving an inequality where the function $f(x)$ is given by the square root of a cubic polynomial. The steps to solve such an inequality typically involve:
Setting the Function Equal to Zero: This is the starting point for solving the equation, as the square root of a number is zero only when the number itself is zero.
Squaring Both Sides: Since the function includes a square root, squaring both sides of the equation gets rid of the radical and simplifies the equation to a polynomial form.
Simplification: After squaring, the equation needs to be simplified. This involves applying exponent rules, such as $(a^m)^n = a^{mn}$, and simplifying any resulting expressions.
Factoring: The resulting polynomial equation is then factored to find the solutions for $x$. This can involve factoring out common terms or using methods like grouping.
Finding Solutions: Each factor set equal to zero gives a potential solution for $x$. These solutions are then tested in the original inequality to determine which are valid.
Presenting Solutions: The final step is to present the solutions in both exact (fractional) form and decimal form, if necessary.
Relevant mathematical concepts include:
- Square roots and radicals
- Exponent rules
- Polynomial factoring (common factor, grouping, etc.)
- Solving linear equations
Understanding these concepts is essential for solving polynomial inequalities and similar algebraic expressions.
