Math

How do you solve a system of nonlinear equations involving a quadratic and a linear equation using the substitution or elimination method?To solve a system of nonlinear equations involving a quadratic and a linear equation using substitution, first solve the linear equation for one variable. Then substitute this expression into the quadratic equation. Solve the resulting quadratic equation for the variable and back-substitute to find the other variable. For elimination, align both equations and eliminate one variable by adding or subtracting the equations, then solve the resulting equation.

How can the Law of Sines be used to determine the radius of the circumscribed circle around a triangle given all three sides of the triangle?To determine the radius (R) of the circumscribed circle around a triangle using the Law of Sines, use the formula R = a / (2 * sin(A)), where ‘a’ is a side of the triangle and ‘A’ is the angle opposite to it. First, find the angles using the sides and the Law of Cosines, then apply the Law of Sines.

How do you find the zeros of a polynomial function using the Rational Root Theorem and synthetic division?To find the zeros of a polynomial function using the Rational Root Theorem and synthetic division, first list all possible rational roots using the theorem. Then, use synthetic division to test each candidate root. If the remainder is zero, the candidate is a root. Repeat the process with the quotient polynomial until all zeros are found.

What are the key differences between linear and quadratic functions?Linear functions have the form f(x) = mx + b, where the graph is a straight line, and the rate of change is constant. Quadratic functions have the form f(x) = ax^2 + bx + c, where the graph is a parabola, and the rate of change varies, creating a curved shape.

How do you solve for x in the equation 3x + 4 = 19?To solve for x, first subtract 4 from both sides of the equation: 3x + 4 – 4 = 19 – 4, which simplifies to 3x = 15. Next, divide both sides by 3: 3x/3 = 15/3, resulting in x = 5.

How do you derive the sine and cosine t-values given an angle on the unit circle, and how are these values used to solve right triangle problems?To derive sine and cosine t-values given an angle θ on the unit circle, identify the coordinates (x, y) where the terminal side of the angle intersects the circle. Here, x = cos(θ) and y = sin(θ). These values can solve right triangle problems by using the definitions of sine and cosine: sin(θ) = opposite/hypotenuse and cos(θ) = adjacent/hypotenuse.

What is the greatest common factor (GCF) of 28 and 42?The greatest common factor (GCF) of 28 and 42 is 14. This is determined by finding the largest integer that divides both numbers without leaving a remainder. In this case, 14 is the highest number that can evenly divide both 28 and 42.

How do I solve a proportion where one of the terms is unknown, like in the equation 3/4 = x/8?To solve the proportion 3/4 = x/8, cross-multiply to get 3 * 8 = 4 * x. This simplifies to 24 = 4x. Divide both sides by 4 to isolate x, resulting in x = 6. Therefore, x equals 6.

How do you interpret the p-value in a hypothesis test?The p-value in a hypothesis test quantifies the probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true. A low p-value indicates strong evidence against the null hypothesis, suggesting it may be false. Common thresholds are 0.05 or 0.01.

What are the differences between perpendicular and parallel lines?Perpendicular lines intersect at a 90-degree angle, forming right angles. Parallel lines, on the other hand, never intersect and remain equidistant from each other at all points. These properties are fundamental in geometry, influencing various mathematical and real-world applications.