Math

How do you solve a system of nonlinear equations involving a quadratic and a logarithmic function?To solve a system of nonlinear equations involving a quadratic and a logarithmic function, you typically use substitution or graphical methods. First, isolate one variable in one equation and substitute into the other. Alternatively, graph both functions and find their intersection points.

If Frank buys 3 notebooks for $3.45 each and a pen for $1.20, what is the total cost of all the items including a 7.5% sales tax?To calculate the total cost, first find the subtotal: 3 notebooks at $3.45 each is $10.35, plus $1.20 for the pen equals $11.55. Adding a 7.5% sales tax ($0.87), the total cost is $12.42.

Define the unit circle in trigonometry

How do you find the vertex of a quadratic function y = ax^2 + bx + c?To find the vertex of a quadratic function y = ax^2 + bx + c, use the formula x = -b/(2a) to find the x-coordinate. Substitute this value back into the original equation to find the y-coordinate. The vertex is at (x, y).

How can you find the values of x for the equation 2x^3 – x^2 + 4x – 1 = 0 using factorization techniques or numerical methods?To solve the equation 2x^3 – x^2 + 4x – 1 = 0, you can use the Rational Root Theorem to test potential rational roots. If factorization is complex, numerical methods like Newton-Raphson or software tools such as MATLAB or Python can approximate the roots efficiently.

What is the distributive property and how can it be used to simplify the expression 4(3 + 2)?The distributive property states that a(b + c) = ab + ac. To simplify 4(3 + 2) using the distributive property, you multiply 4 by each term inside the parentheses: 4 * 3 + 4 * 2, which equals 12 + 8. Therefore, 4(3 + 2) simplifies to 20.

How do you simplify the expression (x^2 – 4)/(x + 2)?To simplify the expression (x^2 – 4)/(x + 2), first recognize that x^2 – 4 is a difference of squares and can be factored into (x + 2)(x – 2). The expression then becomes [(x + 2)(x – 2)] / (x + 2). By cancelling the common factor (x + 2), the simplified expression is x – 2, for x ≠ -2.

How do you solve for x in the equation involving both factors and quadratic expressions: (2x^2 – 4x + 3) = (x – 1)(x^2 + 3x – 2)?To solve for x in the equation (2x^2 – 4x + 3) = (x – 1)(x^2 + 3x – 2), first expand the right-hand side: (x – 1)(x^2 + 3x – 2). Then, equate the expanded form to the left-hand side. Combine like terms to form a single quadratic equation. Finally, solve the quadratic equation using the quadratic formula or factoring, if possible.

How can you prove that the sum of the angles of a cyclic quadrilateral is 360 degrees using properties of inscribed angles?To prove that the sum of the angles of a cyclic quadrilateral is 360 degrees using properties of inscribed angles, consider a cyclic quadrilateral ABCD inscribed in a circle. The opposite angles of a cyclic quadrilateral are supplementary, meaning that ∠A + ∠C = 180° and ∠B + ∠D = 180°. Adding these equations gives (∠A + ∠C) + (∠B + ∠D) = 360°. Therefore, the sum of the angles of a cyclic quadrilateral is 360 degrees.

How do you find the domain and range of a function given its equation?To find the domain of a function, identify all possible input values (x) that will produce a valid output. Check for restrictions like division by zero or negative square roots. To find the range, determine all possible output values (y) by analyzing the function’s behavior and graph. Consider any constraints on y-values.