# Lesson 4 All Things Being Equal Develop Understanding

## Learning Focus

Solve quadratic equations graphically and algebraically.

Make connections between solving quadratic equations and graphing quadratic functions.

How can we use graphs to solve quadratic equations?

## Open Up the Math: Launch, Explore, Discuss

As we have learned, quadratic functions can be very useful models for a lot of real situations. They are part of understanding the motion of objects, business and economic models, and many other applications. Using quadratic functions often requires solving a quadratic equation. This can be a pretty straightforward process, or it can be a little complicated. The good news is that we know a lot about quadratic functions that we can apply to solving equations. Let’s get started.

### 1.

If you were given the quadratic equation,

### 2.

Graph the function:

#### a.

#### b.

For what values of

### 3.

Given the equation

#### a.

Solve the equation algebraically.

#### b.

Graph the function

### 4.

Given the equation:

#### a.

Graph the function

#### b.

Solve the equation algebraically.

Don’t feel bad if you couldn’t solve the equation algebraically. It’s one of those complicated equations that can’t be solved directly with inverse operations. Knowing that the graph of an equation is the set of all of its solutions, we can solve any quadratic equation graphically using technology. Now we’re going to use our understanding of the graph of a quadratic function and its symmetry to develop other algebraic techniques for solving quadratic equations.

Let’s start in a familiar place.

### 5.

If you’re given the function

### 6.

How could you use this kind of thinking to solve:

### 7.

Now you can try a few. Solve each quadratic equation by factoring.

#### a.

#### b.

#### c.

When we were graphing, we saw that some functions are easier to factor, and others are easier to complete the square. Let’s start with one where the work has been done for us so that we can start seeing some relationships.

### 8.

Consider the function:

#### a.

Graph the function:

#### b.

Line of symmetry:

#### c.

Vertex:

#### d.

What are the

#### e.

How far are the

#### f.

Use inverse operations to solve the equation

#### g.

How does solving the equation relate to the graph and the

Don’t be mad, but that equation would have factored easily if it were given in standard form. That’s why the solutions were integers. Not all equations factor easily, and the relationship that we are beginning to see can be very helpful.

Let’s look at a function and equation that do not factor easily from standard form, and see how vertex form can help us solve the equation or find the

### 9.

Start with the function

#### a.

Graph the function by putting it in vertex form.

#### b.

Vertex:

#### c.

Line of symmetry:

#### d.

What do you estimate the

#### e.

What do you estimate the distance from each

#### f.

Starting with

#### g.

How do your solutions compare with your estimates of the

### 10.

Given:

#### a.

Solve the equation by completing the square and using inverse operations.

#### b.

Let

#### c.

What is the distance from line of symmetry to an

### 11.

Given:

#### a.

Solve the equation by completing the square and using inverse operations.

#### b.

Let

#### c.

What is the distance from line of symmetry to an

## Ready for More?

Find two methods for solving this equation graphically:

## Takeaways

Solving quadratic equations using inverse operations:

Example | Procedure |
---|---|

Given | |

Solving quadratic equations by graphing:

Example | Procedure |
---|---|

Given | |

Solving quadratic equations by factoring:

Example | Procedure |
---|---|

Given – The equation must be equal to 0. Rearrange the terms if necessary. | |

Solving quadratic equations by completing the square:

Example | Procedure |
---|---|

Given – Arrange the terms so the equation equals 0, if it isn’t already. | |

## Lesson Summary

In this lesson, we learned methods for solving quadratic equations. Some quadratic equations can be solved using inverse operations and taking the square root of both sides of the equations. Some quadratic equations can be solved by factoring and using the zero product property. Some quadratic equations can be solved by completing the square and then using inverse operations. Quadratic equations that have real solutions can also be solved by graphing, and each of these algebraic methods has connections to graphing.

### 1.

Find the features for the function represented in the graph.

Intervals of Increase:

Intervals of Decrease:

Maximum:

Minimum:

Domain:

Range :

### 2.

Use the functions to find the indicated values.