Basic Algebra

Step 1: Given equation.
The given equation is

Step 2: Simplify the equation.
Using the logarithmic identity , we can rewrite the equation as: Now, use the property to obtain:

Step 3: Equate the arguments.
Since the logarithms are equal, their arguments must also be equal. Thus: Multiplying both sides by 3:

Step 4: Find the value of .
Now, square both sides of the equation to obtain: Expanding the left-hand side: Rearranging the terms: Now, divide both sides by : Simplifying:

Final Answer: 7.

Step 1: Polynomial Degree.
The given polynomial is: The degree of the polynomial is determined by the highest power of , which is 11. Hence, the polynomial is of degree 11.

Step 2: Maximum Number of Zeros.
According to the Fundamental Theorem of Algebra, a polynomial of degree has at most real or complex roots (zeros). Therefore, the polynomial can have at most 11 zeros.

Step 3: Conclusion.
Since the degree of the polynomial is 11, the number of zeros is at most 11. Therefore, the number of zeros is 11.

Final Answer: 11.

Step 1: Understand the terms.
- Mean: The average of the numbers.
- Median: The middle number in an ordered data set.
- Mode: The most frequent number in a data set.
- Range: The difference between the highest and lowest values in the data.

Step 2: Match with the values.
Given the numbers 5, 8, 14, 7, 9, 11, 5, we calculate:
- Mean:
- Median: The middle value of the sorted list (5, 5, 7, 8, 9, 11, 14) is 9.
- Mode: The most frequent value is 5.
- Range:
Thus, the correct matching is .

Final Answer: i - d, ii - a, iii - c, iv - b.

Step 1: Rewriting the equation.
The given equation is . Let , so the equation becomes:

Step 2: Solving the quadratic equation.
Now, solve the quadratic equation by factoring: So, or .

Step 3: Substituting .
Since , we substitute back to find the roots of :
- For , , so or .
- For , , so or .

Step 4: Conclusion.
Therefore, the roots of the equation are .

Final Answer: 2, -1, -2, 1.

Step 1: Defining variables.
Let and . The given equations then become:

Step 2: Solving the system of equations.
We can multiply equation (A) by 2 and subtract from equation (B) to eliminate . Multiplying (A) by 2: Now subtract equation (B) from equation (C): Thus,

Step 3: Substituting in equation (A).
Substitute in equation (A): Thus,

Step 4: Finding .
We now have and . Solve these two equations: 1)
2) Add these equations: Substitute into : Finally, calculate :

Final Answer: 41.

Step 1: General form of the quadratic equation.
The given quadratic equation is: Expanding the equation:

Step 2: Condition for equal roots.
For a quadratic equation to have equal roots, the discriminant must be zero. The discriminant is given by: Here, , , and . Substituting these values into the discriminant formula: For the roots to be equal, , so:

Step 3: Solve for .
Factorizing the equation: Thus, or . Since , we conclude that:

Final Answer: 6.