Algebra 2 Semester B Exam Part 1
1. Solve the following system of equations:
2x + 3y = 10
4x - y = 5
First, solve for y in the second equation:
y = 4x - 5
Substitute this expression for y into the first equation:
2x + 3(4x - 5) = 10
2x + 12x - 15 = 10
14x - 15 = 10
14x = 25
x = 25/14
Now, substitute x back into the second equation to solve for y:
4(25/14) - y = 5
100/14 - y = 5
y = 100/14 - 5
y = 10/7 - 5
y = -25/7
Therefore, the solution to the system of equations is x = 25/14 and y = -25/7.
2. Simplify the following expression:
(3x^2 - 2x + 5)(4x + 1)
Use the distributive property to simplify:
(3x^2 - 2x + 5)(4x + 1)
= 3x^2(4x) + 3x^2(1) - 2x(4x) - 2x(1) + 5(4x) + 5(1)
= 12x^3 + 3x^2 - 8x^2 - 2x + 20x + 5
= 12x^3 - 5x^2 + 18x + 5
Therefore, the simplified expression is 12x^3 - 5x^2 + 18x + 5.
3. Find the domain of the following function:
f(x) = sqrt(x^2 - 4)
The square root function is defined for non-negative values, so the expression inside the square root must be greater than or equal to zero:
x^2 - 4 ≥ 0
x^2 ≥ 4
x ≥ -2 or x ≤ 2
Therefore, the domain of the function f(x) is all real numbers such that x is greater than or equal to -2 and less than or equal to 2.