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# Quadratic Equations and Inequalities Study Guide: GED Math

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Updated on Mar 23, 2011

Practice problems for these concepts can be found at:

Algebra and Functions Practice Problems: GED Math

A quadratic expression is an expression that contains an x2 term. The expressions x2 – 4 and x2 + 3x + 2 are two examples of quadratic expressions. A quadratic equation is a quadratic expression set equal to a value. The equation x2 + 3x + 2 = 0 is a quadratic equation.

Quadratic equations have two solutions. To solve a quadratic equation, combine like terms and place all terms on one side of the equal sign, so that the quadratic is equal to 0. Then, factor the quadratic and find the values of x that make each factor equal to 0. The values that solve a quadratic are the roots of the quadratic.

Example

Solve x2 + 5x + 2x + 10 = 0.

Combine like terms: x2 + 7x + 10 = 0.

Factor: (x + 5)(x + 2) = 0, so x + 5 = 0 and/or x + 2 = 0.

x = –5 and/or x = –2.

–5 + 5 = 0 and –2 + 2 = 0

Therefore, x is equal to both –5 and –2.

### Inequalities

Linear inequalities are solved in much the same way as simple equations. The most important difference is that when an inequality is multiplied or divided by a negative number, the inequality symbol changes direction.

Example

10 > 5; but if you multiply by –3, –30 < –15.

### Solving Linear Inequalities

To solve a linear inequality, isolate the letter and solve the same as you would in a first-degree equation (an equation in which x is raised only to the first power). Remember to reverse the direction of the inequality sign if you divide or multiply both sides of the equation by a negative number.

Example

If 7 – 2x > 21, find x.

Isolate the variable:

7 – 2x > 21

7 – 2x – 7 > 21 – 7

–2x > 14

Then divide both sides by –2. Because you are dividing by a negative number, the inequality symbol changes direction: becomes x < –7, so the answer consists of all real numbers less than –7.

### Functions

A function is an equation with one input (variable) in which each unique input value yields no more than one output. The set of elements that make up the possible inputs of a function is the domain of the function. The set of elements that make up the possible outputs of a function is the range of the function.

A function commonly takes the form f(x) = x + c, where x is a variable and c is a constant. The values for x are the domain of this function. The values of f(x) are the range of the function.

If f(x) = 5x + 2, what is f(3)?

To find the value of a function given an input, substitute the given input for the variable: f(3) = 5(3) + 2 = 15 + 2 = 17.

### Domain

The function f(x) = 3x has a domain of all real numbers. Any real number can be substituted for x in the equation, and the value of the function will be a real number.

The function f(x) = – 4 has a domain of all real numbers excluding 4. If x = 4, the value of the function would be , which is undefined. In a function, the values that make a part of the function undefined are the values that are NOT in the domain of the function.

What is the domain of the function f(x) = ?

The square root of a negative number is an imaginary number, so the value of x must not be less than 0. Therefore, the domain of the function is x ≥ 0.

### Range

As you just saw, the function f(x) = 3x has a domain of all real numbers. If any real number can be substituted for x, 3x can yield any real number. The range of this function is also all real numbers.

Although the domain of the function f(x) = – 4 is all real numbers excluding 4, the range of the function is all real numbers excluding 0, because no value for x can make f(x) = 0.

What is the range of the function f(x) = ?

You already found the domain of the function to be x ≥ 0. For all values of x greater than or equal to 0, the function will return values greater than or equal to 0.

### Nested Functions

Given the definitions of two functions, you can find the result of one function (given a value) and place it directly into another function. For example, if f(x) = 5x + 2 and g(x) = –2x, what is f [g(x)] when x = 3?

Begin with the innermost function: find g(x) when x = 3. In other words, find g(3). Then, substitute the result of that function for x in f(x): g(3) = –2(3) = –6, f(–6) = 5(–6) + 2 = –30 + 2 = –28. Therefore, f [g(x)] = –28 when x = 3.

What is the value of g[f(x)] when x = 3?

Start with the innermost function; this time, it is f(x): f(3) = 5(3) + 2 = 15 + 2 = 17. Now, substitute 17 for x in g(x): g(17) = –2(17) = –34. When x = 3, f [g(x)] = –28 and g[f(x)] = –34.

Practice problems for these concepts can be found at:

Algebra and Functions Practice Problems: GED Math