Algebra II Worksheets

Algebra II worksheets offer a comprehensive and structured way to practice and reinforce key concepts for high school students studying advanced algebra. These worksheets serve as invaluable tools to strengthen one's understanding of complex mathematical equations and provide ample opportunity to hone problem-solving skills.

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Solve for x: 3x + 5 = 17.

To solve for x in the equation 3x + 5 = 17, we first subtract 5 from both sides to isolate the variable. This gives us 3x = 12. Then, divide both sides by 3 to solve for x. Therefore, x = 4.

Simplify the expression: (2x^2 - 3y) + (4x^2 + 5y).

The simplified expression is 6x^2 + 2y.

Factor the equation: x^2 + 5x + 6 = 0.

To factor the equation x^2 + 5x + 6 = 0, we need to find two numbers that multiply to 6 and add up to 5. These numbers are 2 and 3. So, the factored form of the equation is (x + 2)(x + 3) = 0.

Find the vertex of the quadratic function: f(x) = 2x^2 + 8x - 5.

To find the vertex of the quadratic function f(x) = 2x^2 + 8x - 5, we can use the formula for finding the x-coordinate of the vertex, which is given by x = -b/(2a), where a is the coefficient of x^2 and b is the coefficient of x. In this case, a = 2 and b = 8. Plugging these values into the formula, we get x = -8/(2*2) = -2. Then, substitute x = -2 back into the equation to find the y-coordinate: f(-2) = 2*(-2)^2 + 8*(-2) - 5 = 8 - 16 - 5 = -13. Therefore, the vertex of the quadratic function f(x) = 2x^2 + 8x - 5 is (-2, -13).

Solve the system of equations: 2x + 3y = 7 and 4x - 5y = 9.

To solve the system of equations 2x + 3y = 7 and 4x - 5y = 9, we can start by multiplying the first equation by 2 to get 4x + 6y = 14. Then, we can add the second equation to this new equation to eliminate x: (4x + 6y) + (4x - 5y) = 14 + 9. This simplifies to 10y = 23, so y = 23/10. Substituting this value back into the first equation, we get 2x + 3(23/10) = 7, which simplifies to x = 11/10. Therefore, the solution to the system of equations is x = 11/10 and y = 23/10.

Simplify the rational expression: (x^2 - 9)/(x + 3).

To simplify the rational expression (x^2 - 9)/(x + 3), first factor the numerator as the difference of squares: (x^2 - 9) = (x + 3)(x - 3). Now rewrite the expression as ((x + 3)(x - 3))/(x + 3). The (x + 3) terms will cancel out leaving x - 3 as the simplified form of the rational expression.

Find the domain of the function: f(x) = square root of (x - 3).

For the function f(x) = ?(x - 3), the domain includes all real numbers greater than or equal to 3 since the square root of a negative number is not a real number, and considering the square root function, the radicand (expression inside the square root) should be greater than or equal to zero. Therefore, the domain of the function is x ? 3.

Use synthetic division to divide f(x) = 3x^3 + 2x^2 - 5x + 1 by g(x) = x + 2.

Performing synthetic division by using the divisor x + 2, place the coefficients of f(x) = 3x^3 + 2x^2 - 5x + 1, which are 3, 2, -5, and 1, respectively. In the synthetic division grid, bring down the first coefficient, which is 3, multiply it by the divisor x + 2 to get 3, then add it to the next coefficient, giving 5. Multiply this sum by x + 2 to get 10, and continue the process until reaching the last coefficient. The resultant quotient obtained will be the solution after completing the synthetic division process.

Solve the inequality: 2x - 5 > 8x + 3.

To solve the inequality 2x - 5 > 8x + 3, we need to isolate x on one side of the inequality. First, subtract 2x from both sides to get -5 > 6x + 3. Then subtract 3 from both sides to get -8 > 6x. Finally, divide by 6 to get x < -8/6 or x < -4/3, which is the solution to the given inequality.

Find the inverse of the function: f(x) = 4x - 2.

To find the inverse of the function f(x) = 4x - 2, we switch the roles of x and y and solve for y. Therefore, x = 4y - 2. Adding 2 to both sides gives x + 2 = 4y. Dividing by 4 yields y = (x + 2) / 4. Thus, the inverse function is f^-1(x) = (x + 2) / 4.