# Quadratic Equations and Functions

## Solve Quadratic Inequalities

We have learned how to solve linear inequalities and rational inequalities previously. Some of the techniques we used to solve them were the same and some were different.
We will now learn to solve inequalities that have a quadratic expression. We will use some of the techniques from solving linear and rational inequalities as well as quadratic equations.
We will solve quadratic inequalities two ways—both graphically and algebraically.

### Solve Quadratic Inequalities Graphically

A quadratic equation is in standard form when written as ax2 + bx + c = 0. If we replace the equal sign with an inequality sign, we have a quadratic inequality in standard form.
The graph of a quadratic function f(x) = ax2 + bx + c = 0 is a parabola. When we ask when is ax2 + bx + c < 0, we are asking when is f(x) < 0. We want to know when the parabola is below the x-axis.
When we ask when is ax2 + bx + c > 0, we are asking when is f(x) > 0. We want to know when the parabola is above the x-axis.
We list the steps to take to solve a quadratic inequality graphically.
In the last example, the parabola opened upward and in the next example, it opens downward. In both cases, we are looking for the part of the parabola that is below the x-axis but note how the position of the parabola affects the solution.

### Solve Quadratic Inequalities Algebraically

The algebraic method we will use is very similar to the method we used to solve rational inequalities. We will find the critical points for the inequality, which will be the solutions to the related quadratic equation. Remember a polynomial expression can change signs only where the expression is zero.
We will use the critical points to divide the number line into intervals and then determine whether the quadratic expression willl be postive or negative in the interval. We then determine the solution for the inequality.
In this example, since the expression  factors nicely, we can also find the sign in each interval much like we did when we solved rational inequalities. We find the sign of each of the factors, and then the sign of the product. Our number line would like this:
The result is the same as we found using the other method.
We summarize the steps here.
The solutions of the quadratic inequalities in each of the previous examples, were either an interval or the union of two intervals. This resulted from the fact that, in each case we found two solutions to the corresponding quadratic equation ax2 + bx + c = 0. These two solutions then gave us either the two x-intercepts for the graph or the two critical points to divide the number line into intervals.
This correlates to our previous discussion of the number and type of solutions to a quadratic equation using the discriminant.
For a quadratic equation of the form ax2 + bx + c = 0, 
The last row of the table shows us when the parabolas never intersect the x-axis. Using the Quadratic Formula to solve the quadratic equation, the radicand is a negative. We get two complex solutions.
In the next example, the quadratic inequality solutions will result from the solution of the quadratic equation being complex.

### Key Concepts

1. Solve a Quadratic Inequality Graphically


2. How to Solve a Quadratic Inequality Algebraically



### Section Exercises


### Practice Makes Perfect

Solve Quadratic Inequalities Graphically
In the following exercises, ⓐ solve graphically and ⓑ write the solution in interval notation.
In the following exercises, solve each inequality algebraically and write any solution in interval notation.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

### Chapter Review Exercises


### Solve Quadratic Equations Using the Square Root Property

Solve Quadratic Equations of the form 
In the following exercises, solve using the Square Root Property.
Solve Quadratic Equations of the Form 
In the following exercises, solve using the Square Root Property.

### Solve Quadratic Equations by Completing the Square

Solve Quadratic Equations Using Completing the Square
In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
In the following exercises, solve by completing the square.
Solve Quadratic Equations of the form 
In the following exercises, solve by completing the square.

### Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.
Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation
In the following exercises, determine the number of solutions for each quadratic equation.
Identify the Most Appropriate Method to Use to Solve a Quadratic Equation
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

### Solve Equations in Quadratic Form

Solve Equations in Quadratic Form
In the following exercises, solve.

### Solve Applications of Quadratic Equations

Solve Applications Modeled by Quadratic Equations
In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

### Graph Quadratic Functions Using Properties

Recognize the Graph of a Quadratic Function
In the following exercises, graph by plotting point.
In the following exercises, determine if the following parabolas open up or down.
Find the Axis of Symmetry and Vertex of a Parabola
In the following exercises, find ⓐ the equation of the axis of symmetry and ⓑ the vertex.
Find the Intercepts of a Parabola
In the following exercises, find the x- and y-intercepts.
Graph Quadratic Functions Using Properties
In the following exercises, graph by using its properties.
Solve Maximum and Minimum Applications
In the following exercises, find the minimum or maximum value.
In the following exercises, solve. Rounding answers to the nearest tenth.

### Graph Quadratic Functions Using Transformations

Graph Quadratic Functions of the form 
In the following exercises, graph each function using a vertical shift.
In the following exercises, graph each function using a horizontal shift.
In the following exercises, graph each function using transformations.
Graph Quadratic Functions of the form 
In the following exercises, graph each function.
Graph Quadratic Functions Using Transformations
In the following exercises, rewrite each function in the  form by completing the square.
In the following exercises, ⓐ rewrite each function in  form and ⓑ graph it by using transformations.
In the following exercises, ⓐ rewrite each function in  form and ⓑ graph it using properties.
Find a Quadratic Function from its Graph
In the following exercises, write the quadratic function in  form.

### Solve Quadratic Inequalities

Solve Quadratic Inequalities Graphically
In the following exercises, solve graphically and write the solution in interval notation.
In the following exercises, solve each inequality algebraically and write any solution in interval notation.

### Practice Test

Solve the following quadratic equations. Use any method.
Use the discriminant to determine the number and type of solutions of each quadratic equation.
Solve each equation.
For each parabola, find ⓐ which direction it opens, ⓑ the equation of the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and e) the maximum or minimum value.
Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.
In the following exercises, graph each function using transformations.
In the following exercises, solve each inequality algebraically and write any solution in interval notation.
Model the situation with a quadratic equation and solve by any method.