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# Evaluate|Definition & Meaning

## Definition

In **Mathematics,** the word **“evaluate”** means to find the value of a **specific equation,** or **expression** and calculate the solution to the problem. The expression “to evaluate” is usually used with **algebraic expressions** or other equations to find the resulting solution to the problem that is given in the expression, such as, finding the **missing number** of an **expression** or finding the value of **x** in a given algebraic equation.

**Expressions,** such as simple **mathematical operations** between two numbers, always result in a **numerical value** that gives us the meaning of the expression in general. Thus, we can also say that finding the numerical value of an expression is mathematically said to **evaluate** the given expression.

Furthermore, expressions that include an **algebraic variable** also are evaluated to find the **numerical value** of the algebraic variable in the expression. This way, **multiple problems** are simplified into simple algebraic equations and expressions for ease of **understanding** and easy **solving** and evaluating of such problems.

Moreover, it is important to note that expressions or equations that are being evaluated must result in a **constant numerical value.** If the expression has a **variable** in it then this variable must be **substituted** by a **constant value** to evaluate the expression.

## Simplify, Solve and Evaluate

In mathematics, explaining** problems** and what to find or calculate is usually done by using these **three words** or phrases: **Evaluate, Solve,** and **Simplify.** These words are most commonly used in exercises when finding solutions to various types of problems or just re-writing the expressions or problems in a way to easily understand the problem for which we need to find the solution.

As explained in the last section, **evaluation** is usually used to find a **numerical value** of an expression or an equation with no variables or equations as the result. The answer must always consist of a **numerical value.** An example of such type is to **evaluate** a **given expression** $2+2$ and by evaluating it, we obtain the answer **four** which is a **numerical solution.**

The word **“solve”** is usually used to **explain** the **solution** by **isolating** a **particular variable** in an equation such as those in **quadratic equations.** Here, we solve the quadratic equations for finding the value of **x** that satisfies the given equation. Furthermore, when using the word solve**,** it can **either** mean the **solution** is in the **form** of a **variable** or a **constant value.**

Moreover, solving an **equation** can also be done using **graphical expressions** to find the resulting **curve equation** or, for example, the **roots** of a **curve** given as a graph.

Finally, **simplify** is used to **rewrite complicated equations** or expressions into **easier alternate versions** of the expressions that will help in finding the solution to the problem more easily. This simplification can be explained by **simplifying** quadratic equations using **different techniques** to rewrite them into simpler multiple forms.

## Examples Featuring Evaluations of Expressions and Equations

### Example 1

An **equation** is given below:

y = 5x$^2$ + 3x – 3

**Simplify** this **expression** and **evaluate** this when the expression is equal to **zero**

### Solution

To simplify this expression, we utilize the **completing square method** to rewrite the equation into an easier expression.

y = 5x$^2$ + 3x – 3

y = 5$\left(x^2+\dfrac{3}{5}x-\dfrac{3}{5}\right)$

y = $5\left(x^2+\dfrac{3}{5}x-\dfrac{3}{5}+\left(\dfrac{3}{10}\right)^2-\left(\dfrac{3}{10}\right)^2\right)$

y = $5\left(\left(x+\dfrac{3}{10}\right)^2-\dfrac{3}{5}-\left(\dfrac{3}{10}\right)^2\right)$

y = $5\left(\left(x+\dfrac{3}{10}\right)^2-\dfrac{69}{100}\right)$

y = $5\left(x+\dfrac{3}{10}\right)^2-\dfrac{69}{20}$

Thus we rewrite this expression into a simpler form.

Now we will **evaluate** this expression with the value of **y** equal to **zero.**

$5\left(x + \dfrac{3}{10}\right)^2-\dfrac{69}{20} = 0$

$5\left(x + \dfrac{3}{10}\right)^2 = \dfrac{69}{20}$

$ \left(x + \dfrac{3}{10}\right)^2 = \dfrac{69}{100}$

$x + \dfrac{3}{10} = \sqrt{\dfrac{69}{100}} $

x = $- \dfrac{3}{10} \pm \dfrac{\sqrt{69}}{10}$

Thus the above **value of x** is evaluated to be either **0.531** or** -1.131**.

### Example 2

A **quadratic equation** is given such that:

x$^2$ + 2x – 3 = 0

Evaluate the **roots** of the **variable** **x** in this equation by first **simplifying** the equation into a simpler form. Also, **draw** a **graph** to further confirm that the roots are **verified.**

### Solution

For this example, we are given a quadratic equation that can be easily **simplified** into an **alternate expression** by using the **completing square method.** This way, the solution can be easily found with fewer steps at the end.

x$^2$ + 2x – 3 = 0

x$^2$ + 2x + 1$^2$ – 3 – 1$^2$ = 0

(x + 1)$^2$ – 3 – 1 = 0

(x + 1)$^2$ – 4 = 0

(x + 1)$^2$ = 4

x + 1 = $\pm\sqrt{4}$

x + 1 =$ \pm$ 2

x = – 1 $\pm$ 2

Thus the above** value of x** is evaluated to be either **-3** or **1.**

Furthermore, the above value can be verified by drawing the **graph** of the **quadratic equation** and finding the **intersecting points** on the **x-axis** that are the **roots** of the **x**.

### Example 3

An **algebraic equation** is given below:

4x + 3x + 9 = 33

**Evaluate** the above algebraic equation to find the **value of x**

### Solution

This is a fairly **simple equation** to **evaluate** and the following is the process to evaluate the value of **x**.

4x + 3x + 9 = 33

7x = 33 – 9

7x = 24

x = $\dfrac{24}{7}$

Hence, the **equation** is evaluated to find the value of** x** being **24/7** or **3.42**

*All drawings and mathematical graphs are made using GeoGebra.*