To solve the linear equation 12 + 3b = 15b, we need to isolate the variable b. Here are the steps to find the solution: Start with the original equation: 12 + 3b = 15b Next, we want to collect all the terms containing b on one side of the equation. To do this, subtract … Read more
Finding the Factors of the Polynomial The polynomial in question is f(x) = 5x³ – 24x² – 75x + 14. To determine the factors of this polynomial, we can use several methods including synthetic division, factoring by grouping, or applying the Rational Root Theorem. Steps to Factor the Polynomial: Identify Rational Roots: Using the Rational … Read more
When dealing with quadratic equations, which are typically expressed in the standard form ax2 + bx + c = 0, the discriminant plays a crucial role in determining the nature of the solutions. The discriminant is calculated using the formula D = b2 – 4ac. If the discriminant (D) equals zero, it indicates that the … Read more
To solve the compound inequality 22 < 5x < 7 + 3, we first simplify the right side of the inequality. The expression 7 + 3 equals 10, so we can rewrite the original compound inequality as: 22 < 5x < 10 Next, we will separate this compound inequality into two parts to make it ... Read more
To find the points on the ellipse defined by the equation 4x² + y² = 4 that are farthest from the point (1, 0), we can follow a systematic approach that involves both geometric intuition and optimization techniques. Step 1: Understand the Ellipse The ellipse represented by the equation can be rewritten in standard form: … Read more
To find the length of line segment AB, we first need to understand what is meant by the coordinates given: AC(4, AE(7)) and AD(10). Assuming AC(4, 7) represents the coordinates of point A, we take point A as (4, 7). Next, we interpret AD(10) in this context. However, as it’s stated, it seems a bit … Read more
To find the approximate amount of syrup in 45 gulab jamuns, we first need to determine the volume of a single gulab jamun, which can be approximated as a cylinder topped with two hemispherical ends. The formula for the volume of a cylinder is: Vcylinder = πr2h And the formula for the volume of a … Read more
To determine if the points (3, 6), (2, 9), and (0, 4) are collinear, we need to check if the slopes between each pair of points are equal. 1. **Find the slope between the first two points (3, 6) and (2, 9):** The formula for slope (m) between two points (x1, y1) and (x2, y2) … Read more
Understanding the Graph Behavior of f(x) = 2×3 + 26x + 24 The behavior of the graph of a polynomial function, such as f(x) = 2×3 + 26x + 24, can be analyzed by examining various characteristics including: 1. End Behavior As x approaches +∞ (positive infinity), f(x) approaches +∞. As x approaches -∞ (negative … Read more
To find a third degree polynomial equation with rational coefficients that has the roots 1 and 3i, we first need to remember that if a polynomial has real coefficients, any non-real roots must occur in conjugate pairs. Therefore, if 3i is a root, the conjugate -3i must also be a root. Given the roots, the … Read more