Question

shows "the sum of the square of a number and three less than the number." $$ \begin{array}{|l|l|} \hline \text { Select... } & \nabla \\ \hline x^2+x-3 \\ \hline x^2-3 x \\ \hline 3-x+x^2 \\ \hline \end{array} $$

Short answer

The correct expression is \( x^2 + x - 3 \).

Step-by-step solution

Define Variables

Let the number be represented by the variable x.

Interpret 'the square of a number'

The square of the number x can be written as \( x^2 \).

Interpret 'three less than the number'

Three less than the number x is written as \( x - 3 \).

Sum the Expressions

The sum of the square of the number and three less than the number is \( x^2 + (x - 3) \).

Simplify the Expression

Simplify the expression: \( x^2 + x - 3 \). This is our final expression.

Verify the Expression in the Options

Check the multiple-choice options provided. The expression that matches our result is \( x^2 + x - 3 \).

Key concepts

Variable Definition

In algebra, a variable is a symbol that represents a number we don't know yet. Using variables allows us to create general formulas and solve a wide range of problems.
For this problem, we need to find the sum of two expressions involving the same unknown number. Let's use the variable \( x \) to represent this unknown number.
By defining \( x \) as our number, we can then build mathematical expressions around it.
For example, if the number we are interested in is 7, then \( x \) would be equal to 7. If the number is 42, \( x \) would be 42. The power of using variables lies in handling unknown or generalized cases.

Expression Simplification

Expression simplification means reducing a math expression to its simplest form.
This involves combining like terms, performing basic arithmetic, and sometimes factoring.
In our problem, we simplify the sum of two expressions: the square of a number, \( x^2 \), and three less than the number, \( x-3 \). When combined, we get: \[ x^2 + (x - 3) \].
After removing the parentheses and combining like terms, the expression is simplified to \( x^2 + x - 3 \).

Quadratic Equations

Quadratic equations are second-degree polynomial equations in one variable.
They take the form \[ ax^2 + bx + c = 0 \], where \( a, b, \) and \( c \) are constants.
In this problem, our simplified expression \[ x^2 + x - 3 \] is a quadratic expression, not a complete quadratic equation.
However, if we were to set this expression equal to zero, \ x^2 + x - 3 = 0 \, it becomes a quadratic equation that we could solve to find the values of \( x \).

Algebraic Interpretation

Algebraic interpretation involves translating word problems into algebraic expressions.
This means understanding keywords like 'sum,' 'difference,' 'product,' and 'quotient.'
In this exercise, the phrase 'the sum of the square of a number and three less than the number' needs interpretation.
'Square of a number' is \( x^2 \), and 'three less than the number' is \( x - 3 \).
Combining these gives us \ x^2 + (x - 3) \.
This process transforms the worded problem into a form we can manipulate with algebra.