# 11.3: Quadratic Formula

**At Grade**Created by: CK-12

*This activity is intended to supplement Algebra I, Chapter 10, Lesson 5.*

## Problem 1

1. Identify the zeros of \begin{align*}y = x^2 - 4\end{align*} by graphing the equation in \begin{align*}Y=\end{align*}. If needed, use the **zero** command found under \begin{align*}2^{nd}\end{align*} **[CALC]**. Write the zeros below.

2. You may already know the zero product property, and can demonstrate why the following are the solutions to the equation above:

\begin{align*}x + 2 = 0\end{align*} and \begin{align*}x - 2 = 0\end{align*}

3. A program, ** QUAD**, is provided that has the Quadratic Formula defined. Use \begin{align*}A = 1, B = 0\end{align*}, and \begin{align*}C = -4\end{align*}. What are the solutions to the equation \begin{align*}y = x^2 - 4\end{align*}?

## Problem 2

4. Now, examine the graph of \begin{align*}y = x^2 + x - 6\end{align*}. Graph the equation in \begin{align*}Y=\end{align*}. Determine the zeros. Write the factored form below.

Use the ** QUAD** program again. You only need to enter in the correct values for \begin{align*}a, b\end{align*}, and \begin{align*}c\end{align*}. This should confirm your answers for the \begin{align*}x-\end{align*}intercepts.

5. What are the solutions to the equation \begin{align*}y = x^2 + x - 6\end{align*}?

## Problem 3

6. Now, examine the graph of \begin{align*}y = x^2 - 4x + 4\end{align*}. Graph the equation in \begin{align*}Y=\end{align*} and determine the zeros. Write the factored form below.

7. Using the ** QUAD** program, what are the solutions to the equation \begin{align*}y = x^2 - 4x + 4\end{align*}?

## Exercise 4

8. Explore \begin{align*}y = x^2 - 2x - 7\end{align*}, which is not factorable with integers. You may ask why this quadratic function is not factorable and the previous examples were. Make a conjecture about why you think this could be true:

- “Some quadratic equations are not factorable with integers because…”

or

- “Quadratic equations are only factorable with integers when…”

9. Solve the following equations using the ** QUAD** program.

- \begin{align*}y = x^2 - 2x - 7\end{align*}
- \begin{align*}y = -3x + x +3\end{align*}

10. Finally, use **Lists** to calculate the value of the discriminant for the previous two problems, whose solutions were irrational. Enter the \begin{align*}A\end{align*} coefficient in \begin{align*}L_1,\ B\end{align*} in \begin{align*}L_2\end{align*}, and \begin{align*}C\end{align*} in \begin{align*}L_3\end{align*}. Then, in \begin{align*}L_4\end{align*}, move to heading and enter the formula for the discriminant shown at the right.

- \begin{align*}y = x^2 - 2x - 7\end{align*} Solution: __________________
- \begin{align*}y = -3x + x +3\end{align*} Solution: __________________

**Extensions/Homework**

Use the formula in \begin{align*}L_4\end{align*} (above) to calculate the Discriminant for several other quadratics. Decide if the equation is factorable using integers, then solve it. Factor the quadratic if possible, if not, solve by the quadratic formula.

- \begin{align*}y = x^2 - 6x + 9\end{align*}
- \begin{align*}y = 3x^2 + 4x + 5\end{align*}
- \begin{align*}y = -4x^2 + 2x + 2\end{align*}
- \begin{align*}y = 7x^2 + x - 8\end{align*}
- \begin{align*}y = 2x^2 - 5\end{align*}

Look at the flow chart below and discuss with another student how to use it to answer these homework problems.