The sum of two numbers is 40 and their difference is 10. What are the two numbers?
To find the two numbers, let’s denote them as x and y. According to the problem, we have two equations: x + y = 40 (1) x – y = 10 (2) Now, we can solve these equations step by step. First, we can add both equations (1) and (2): (x + y) + (x […]
The sum of two numbers is 19 and their difference is 7. What are the two numbers?
Let’s denote the two numbers as x and y. We are given two equations based on the information provided: The sum of the numbers: x + y = 19 The difference of the numbers: x – y = 7 We can solve these equations step by step. First, let’s rearrange the second equation to express […]
Rewrite fx = 3x² + 22x + 1 from vertex form to standard form
To convert the given quadratic function from vertex form to standard form, we first need to understand what both forms look like. The vertex form of a quadratic function is often written as: f(x) = a(x – h)² + k Where (h, k) is the vertex of the parabola. The standard form is typically presented […]
What is the difference between the smallest 6-digit number and the greatest 4-digit number?
The smallest 6-digit number is 100000, and the greatest 4-digit number is 9999. To find the difference between these two numbers, we simply need to subtract the greatest 4-digit number from the smallest 6-digit number. So, the calculation would look like this: 100000 – 9999 = 90001 Therefore, the difference between the smallest 6-digit number […]
The product of a number and 8 is at least 25
To understand the statement, let’s define the variable. Let the number be represented by x. The phrase “the product of a number and 8” translates mathematically to 8x. The term “at least 25” means that the product must be greater than or equal to 25. Therefore, we can write the inequality as: 8x ≥ 25 […]
How to Solve Equations with Fractions and Variables in the Denominator
Solving equations that involve fractions and variables in the denominator can seem challenging at first, but with a few simple steps, you can tackle them effectively. First, let’s look at an example equation: Example: Solve the equation: \( \frac{x}{2} + \frac{3}{x} = 5 \) 1. **Identify the Problem**: Start by observing the fractions and the […]
Identify and Calculate the Area and Perimeter for Each Quadrilateral
To identify and calculate the area and perimeter of quadrilaterals, we first need to recognize the different types of quadrilaterals: squares, rectangles, parallelograms, trapezoids, and rhombuses. Each type has its own formulas for area and perimeter. 1. Square Area: The area of a square is calculated using the formula: A = s², where s is […]
If x is real, what are the maximum values of the expression x^2 + 14x + 9 + 22x + 3?
To find the maximum value of the expression x^2 + 14x + 9 + 22x + 3, we first simplify it. Combining like terms, we can rewrite the expression as: x^2 + (14x + 22x) + (9 + 3) = x^2 + 36x + 12. This expression is a quadratic equation in the standard form […]
Solve and Graph the Absolute Value Inequality 2x – 4 < 8
To solve the absolute value inequality 2x – 4 < 8, we follow these steps: Start by isolating the absolute value expression. We can do this by adding 4 to both sides: 2x - 4 + 4 < 8 + 4 2x < 12 Next, divide both sides by 2: x < 6 Now we […]
The sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares.
Let the sides of the two squares be a and b. The area of the squares can be expressed as: Area of the first square: a² Area of the second square: b² According to the problem, we have the first equation: a² + b² = 468 (1) The perimeter of a square is given by […]