College Algebra Review

Front

What is the process to solve the equation 3(2x-1)+4=-2(x+4)?

Back

To solve the equation, first distribute the terms: 6x - 3 + 4 = -2x - 8. Combine like terms: 6x + 1 = -2x - 8. Add 2x to both sides: 8x + 1 = -8. Subtract 1 from both sides: 8x = -9. Finally, divide by 8: x = -9/8.

Front

How do you find the vertex of the quadratic function f(x)=x^2-6x+11?

Back

To find the vertex of a quadratic function in the form f(x)=ax^2+bx+c, use the formula x = -b/(2a). Here, a=1 and b=-6, so x = 6/2 = 3. Substitute x back into the function to find y: f(3) = 3^2 - 6(3) + 11 = 2. Thus, the vertex is (3,2).

Front

What is the domain of the function g(x)=√(5x+10)?

Back

The domain of a function is the set of all possible input values (x) that will not cause any mathematical errors. For g(x)=√(5x+10), the expression inside the square root must be non-negative: 5x + 10 ≥ 0. Solving this gives x ≥ -2. Therefore, the domain is [-2,∞).

Front

How do you solve the equation e^(8x)=17 in terms of ln?

Back

To solve e^(8x)=17, take the natural logarithm of both sides: ln(e^(8x)) = ln(17). Using the property of logarithms, this simplifies to 8x = ln(17). Finally, divide by 8 to isolate x: x = ln(17)/8.

Front

What is the solution to the logarithmic equation log_5(x-11)=2?

Back

To solve log_5(x-11)=2, rewrite it in exponential form: x - 11 = 5^2. This simplifies to x - 11 = 25. Adding 11 to both sides gives x = 36.

Front

What is the standard form of a quadratic equation?

Back

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

What is the quadratic formula used for?

Back

The quadratic formula is used to find the solutions (roots) of a quadratic equation in the form ax^2 + bx + c = 0. It is given by x = (-b ± √(b² - 4ac)) / (2a).

Front

What does the discriminant of a quadratic equation tell us?

Back

The discriminant, given by b² - 4ac, indicates the nature of the roots of a quadratic equation. If it is positive, there are two distinct real roots; if zero, there is one real root; if negative, there are two complex roots.

Front

How do you convert a logarithmic equation to exponential form?

Back

To convert a logarithmic equation log_b(a) = c to exponential form, rewrite it as b^c = a, where b is the base, a is the argument, and c is the exponent.

Front

What is the vertex form of a quadratic function?

Back

The vertex form of a quadratic function is f(x) = a(x-h)² + k, where (h,k) is the vertex of the parabola.