*b*^2 - 4

*ac*, is called the discriminant and can be used to determine the number and type of solutions to the quadratic equation without doing all the work to find the actual solutions.

**Use the Discriminant to determine the number and type the solutions**.

**Solve**.

**Geometry Problem**: The area of a square is 32 square feet. Find the length of each side of the square.

**Geometry Proble**m: The area of a circle is square centimeters. Find the length of the diameter.

**Geometry Problem**: The length of a rectangle is 2 less than 3 times its width. If the area measures 65 square meters find the dimensions of the rectangle.

**Ti**p: Be sure to include the proper units in your answer. Most instructors will mark off a few points if you neglect the units.

**Geometry Problem**: The base of a triangle measures twice that of the height. If the area of the triangle is 25 square inches find the length of the base and height.

**Pythagorean Theorem Problem**: The lengths of the legs of a right triangle are 6 inches and 8 inches. Find the length of the hypotenuse.

**Pythagorean Theorem Problem**: The base of a 10 foot ladder is positioned 4 feet from a wall and leaned against it. Determine how high the ladder reaches.

**Pythagorean Theorem Problem**: A computer screen has dimensions 14.5 inches by 11 inches. Find the length of the diagonal.

**Pythagorean Theorem Problem**: If Joe traveled 100 miles south and 50 miles west to the beach, then how far from home is he?

If an object is launched from a height of s feet with an initial speed of feet per second, its height is given by the following formula.

**: An object is thrown from the ground at an initial speed of 32 feet per second. How will it take for the object to come back to the ground?**

Projectile Problem

Projectile Problem

**Projectile Problem**: An object is launched from the top of a 32 foot building at an initial speed of 128 feet per second. How long will it take to reach the ground?

**Projectile Problem**: An object is launched from ground level at a speed of 128 feet per second. How long will it take to reach a height of 256 feet?

**Work-Rate Problem**: It usually takes Joe’s son 2 hours more time working alone to do the weekly yard work. If Joe and his son do the work together it takes 1 1/2 hours. How long would it take Joe to do the yard working alone?

**Video Examples on YouTube**: