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To find the two numbers based on the given conditions, we can set up two equations. Let’s denote the two numbers as x and y. According to the problem, we have: Equation 1 (sum): x + y = 28 Equation 2 (difference): x – y = 4 Now, we can solve these equations step by … Read more

The Rational Root Theorem is a powerful tool used in algebra to identify possible rational roots of polynomial equations. To apply this theorem to the polynomial equation x3 + x2 + x + 3 = 0, we begin by identifying its coefficients and looking at the structure of the polynomial. In our case, the polynomial … Read more

To find the equation of a plane that passes through a specific point and contains a line, we’ll start by determining key components of the problem. We know the plane should pass through the point (0, 7, 7) and it contains the line given by the parametric equations: x = 1 + 3t y = … Read more

Finding Unit Vectors at a 60-Degree Angle To find two unit vectors that form a 60-degree angle with the vector v = (3, 4), we can follow these steps: Normalize the Vector: First, we need to convert vector v into a unit vector by calculating its magnitude. This is done using the formula: Magnitude of … Read more

To solve the equation f(g(x)) = 0, we first need to understand the relationship between the two functions given: f(x) = 16x + 30 g(x) = 14x + 6 First, we will find the expression for g(x): g(x) = 14x + 6 Next, we substitute g(x) into f(x): f(g(x)) = f(14x + 6) = 16(14x … Read more

In triangle XYZ, if MZ, MX, and MY are the segments that connect point M (the intersection point of some lines such as the medians or altitudes) to vertices Z, X, and Y respectively, several important properties can be deduced about triangle XYZ based on these segments: Median Property: If M is the midpoint of … Read more

Finding the Exact Values of the Six Trigonometric Functions for 120 Degrees The six trigonometric functions include sine, cosine, tangent, cosecant, secant, and cotangent. To find their exact values for an angle of 120 degrees, we can use the unit circle concept and reference angles. Step 1: Identify the Reference Angle The angle of 120 … Read more

The function in question is a quadratic function expressed as: y = 2x² – 32x + 12 This function opens upwards since the coefficient of the x² term (2) is positive. To find the maximum or minimum value of this function, we need to determine the vertex of the parabola. The x-coordinate of the vertex … Read more

To determine the other factor of the polynomial a² + 7a + 10 given that (a – 5) is one of its factors, we can perform polynomial division or use the factorization method. First, we can express the polynomial as follows, knowing one of its factors: a² + 7a + 10 = (a – 5)(?) … Read more

The quadratic formula is a powerful tool used in mathematics for solving quadratic equations, which are equations of the form ax² + bx + c = 0, where a, b, and c are constants and a is not equal to zero. This formula provides the solutions, or roots, of the quadratic equation, which can be … Read more