Lesson 12 Formula for Success Solidify Understanding

Jump Start

1.

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

$3 x^{2} - 6 x - 4 = 0$

2.

Given $f (x) = 3 x^{2} - 6 x - 4$ , use your solutions from problem 1 to find:

a.

Line of symmetry:

b.

The distance from the line of symmetry to the $x$ -intercepts:

Learning Focus

Understand and use a formula for solving quadratic equations.

Is there any method that works for solving all quadratic equations?

Open Up the Math: Launch, Explore, Discuss

Solving an equation by completing the square seems like it goes exactly the same way every time. Let’s try to generalize this process by using: $f (x) = a x^{2} + b x + c$ to represent any quadratic function that has $x$ -intercepts and $a x^{2} + b x + c = 0$ .

Start the process the usual way by putting the equation into vertex form. It’s a little tricky, but just do the same thing with $a$ , $b$ , and $c$ as what you did in the last problem with the numbers.

Here are some steps to think about using a diagram:

1.

How many big blocks do you have?

2.

Imagine one of the big blocks and the extra smaller blocks. What goes into each of the sections? You’ve been given a start below. Try filling in the other sections.

Now that you have filled in the small square, what expression do you need to subtract from $c$ ? Don’t forget that you have $a$ big blocks and the diagram shows just one of them. Complete the diagram including the lengths of the sides.

3.

a.

What is the equivalent expression for $f (x) = a x^{2} + b x + c$ in vertex form?

b.

What is the line of symmetry of the parabola?

c.

What is the vertex of the parabola?

4.

Write and solve the equation for $f (x) = 0$ just using inverse operations and some fancy algebra.

Solve each equation using the quadratic formula.

5.

$x^{2} + 5 x + 1 = 0$

6.

$x^{2} - 5 x - 5 = 0$

7.

$x^{2} - 6 x - 3 = 0$

8.

$2 x^{2} + 8 x + 5 = 0$

9.

Now that you’ve tried using the quadratic formula, here are a few questions to consider. Record your answers below.

a.

Why do the solutions from the quadratic formula contain a “ $\pm$ ”?

b.

Is it possible to have only one solution to a quadratic equation? What would that mean about the associated function?

c.

What does it mean if the number inside the root of a solution turns out to be negative?

Ready for More?

Solve this equation using the quadratic formula $2 x^{2} + 6 x + 5 = 0$ .

How do you interpret the solutions?

Examine the graph of the related function after finding the solutions that are undefined as real numbers.

Takeaways

The quadratic formula:

Vocabulary

quadratic formula
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we found a formula by completing the square for solving quadratic equations. We used the quadratic formula to find exact and approximate solutions to quadratic equations and connected those solutions to the graphs of quadratic functions. We learned to rewrite exact solutions so that fractions do not contain common factors in the numerator and denominator and the square roots do not contain factors that are perfect squares.

Retrieval

1.

How do you find $y$ -intercepts for a given equation?

2.

How do you find $x$ -intercepts for a given equation?

Write each of the expressions below in factored form.

3.

$x^{2} - x - 72$

4.

$x^{2} - 5 x - 24$

5.

$x^{2} + 14 x + 13$