Problem 65 - Elementary Algebra

Question

Translate the phrases or sentences to mathematical expressions or equations. A number multiplied by eleven more than itself is six.

Step-by-Step Solution

Answer

Question: Translate the given word problem into a mathematical expression and solve for the unknown number: A number multiplied by eleven more than itself is six. Answer: x * (x + 11) = 6. To solve for 'x', further steps are required to solve the quadratic equation x^2 + 11x - 6 = 0.

Step 1: Identify the unknown

Let 'x' be the unknown number.

Step 2: Write the mathematical expression

Given that, a number multiplied by eleven more than itself is six. So, we can write this as: x * (x + 11) = 6

Step 3: Expand and write the equation

Expanding the expression, we get: x^2 + 11x = 6

Step 4: Write the equation in the standard form

Subtract 6 from both sides to get the equation in standard form for a quadratic equation: x^2 + 11x - 6 = 0 This will be final mathematical equation to represent the given exercise. Students can further solve the quadratic equation to find the value of 'x'.

Key Concepts

Algebraic Expressions

An algebraic expression is like a recipe for mathematical operations, using numbers, variables, and arithmetic operators. Variables represent unknown values and can stand for any number. For instance, consider the phrase 'a number multiplied by eleven more than itself'. To translate this to mathematics, one assigns a variable, e.g., 'x', to represent 'a number'. The 'eleven more than itself' part suggests we are dealing with 'x + 11'. Expression building involves combining variables with constants (like the number 11) and operation signs.

Translating English phrases into algebraic expressions is a vital skill because it helps students solve real-life problems mathematically. Students should remember the order of operations, known as PEMDAS - Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. In our case, multiplication between 'x' and '(x + 11)' is implied, so one must be familiar with algebraic conventions for translating phrases accurately.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation, recognizable by the highest exponent of the variable being two. The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where 'a', 'b', and 'c' are constants, and 'x' represents the variable. Our previously translated phrase led to the equation \( x^2 + 11x = 6 \), which is almost in standard form.

To put this equation in standard form, one must ensure that one side is equal to zero. This is often achieved by moving all terms to one side of the equation. For the given problem, subtracting 6 from both sides yields \( x^2 + 11x - 6 = 0 \), now a proper quadratic equation. Understanding quadratic equations is essential as they appear frequently in various disciplines, from physics to economics, and solving them allows for the analysis of situations involving area, projectile motion, and profit maximization, among many others.

Mathematical Problem Solving

Effective mathematical problem solving requires a systematic approach to dissect and understand the problem before applying mathematical principles to solve it. Start by carefully reading the problem and identifying key pieces of information. Next, choose appropriate mathematical tools. In our case, an algebraic expression effectively represents the phrase provided, which then evolves into a quadratic equation.

Once the proper equation is formed, utilize methods such as factoring, completing the square, or using the quadratic formula to find solutions to the equation. It's crucial to practice with a variety of problems to become comfortable with these techniques and to understand which method suits a particular problem best.