What are the real and complex solutions of the polynomial equation x³ + 8 = 0?
To find the solutions of the polynomial equation x³ + 8 = 0, we start by rewriting the equation in a simpler form: x³ = -8 Next, we can take the cube root of both sides to find the real solution. The cube root of -8 is: x = ∛(-8) = -2 So, x = […]
How to Solve a Quadratic Equation x² + 10x + 24 by Completing the Square
To solve the quadratic equation x² + 10x + 24 = 0 by completing the square, follow these steps: Move the constant term to the other side: Start by rearranging the equation: x² + 10x = -24 Complete the square: To complete the square, we need to add the square of half the coefficient of […]
What is the y intercept of the line with the equation 4x + 2y = 12?
The y-intercept of a line is the point where the line crosses the y-axis. To find this point, we can set the value of x to 0 and solve for y in the given equation. Starting with the equation: 4x + 2y = 12 We substitute x with 0: 4(0) + 2y = 12 This […]
How do you simplify 1 tan2x 1 tan2x?
To simplify the expression 1 tan2x 1 tan2x, we need to clarify it first. It looks like there may be a small formatting issue. It seems you’re asking about the simplification of 1 + tan²(x). Using the Pythagorean identity from trigonometry, we know that 1 + tan²(x) = sec²(x). This identity arises because: tan(x) is […]
What is the antiderivative of e^(2x)?
The antiderivative of e^(2x) can be found using the concept of integration. The function e^(2x) is an exponential function, and when integrating exponential functions, we typically apply the formula for the antiderivative of e^(kx), where k is a constant. In this case, k = 2. The general formula for the antiderivative is: ∫ e^(kx) dx […]
What is the value of the range of the function f(x) = x² – 2?
The function f(x) = x² – 2 is a quadratic function, which means its graph is a parabola. The general form of a quadratic function is f(x) = ax² + bx + c, where ‘a’, ‘b’, and ‘c’ are constants. In this case, ‘a’ is equal to 1 (which is positive), indicating that the parabola […]
Solve for x: 5x + 10 = 30 – 15x
To solve the equation 5x + 10 = 30 – 15x, we need to isolate the variable x. First, let’s get all the x terms on one side of the equation and the constant terms on the other side. We can do this by adding 15x to both sides: 5x + 15x + 10 = […]
How do you simplify the expression 2 divided by (2 + 5i)?
To simplify the expression 2 / (2 + 5i), we can multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a – bi. In this case, the conjugate of 2 + 5i is 2 – 5i. So, let’s rewrite the expression: […]
If angle A has a measurement of 42 and is complementary to angle B, what’s the measure of angle B?
To find the measure of angle B, we need to recall the definition of complementary angles. Two angles are said to be complementary if the sum of their measures is equal to 90 degrees. Given that angle A measures 42 degrees, we can set up the following equation: Angle A + Angle B = 90 […]
What is the image of (1, 6) for a 90 counterclockwise rotation about the origin?
To find the image of the point (1, 6) after a 90-degree counterclockwise rotation about the origin, we can use the rotation formula for points in the Cartesian plane. The formula for rotating a point (x, y) counterclockwise by an angle θ is: (x’, y’) = (x * cos(θ) – y * sin(θ), x * […]