5.6: The Quadratic Formula and the Discriminant

Objectives: Students will be able to…• Solve a quadratic equation using the quadratic

formula• Use the discriminant to determine the number of

solutions a quadratic equation has

The Quadratic Formula!!!

• Works to solve every quadratic equation in standard form: ax2 + bx + c= 0, a ≠ 0

(must set equation equal to 0 first!!)

IT’S MAGIC!!!

a

acbbx

2

42

Solve using the quadratic formula.

3583 2 xx

Now graph the equation and check!! Then solve by factoring!!

Solve using the quadratic formula.

12x-5 = 2x2 +13

Graph it. What do you notice?Solve by factoring. What do you notice?

Solve by the quadratic formula

1. x2 + 3x -2 = 0 2. x2 = 2x – 5

Graph it. What do you notice?

The Discriminant

• Tells us how many solutions a quadratic equation has and the nature of them (real or imaginary)

When you have a quadratic equation in standard form: ax2 + bx + c = 0, the discriminant is:

b2 – 4ac

How to use the discriminantValue of the discriminant

Number and nature of solutions.

What this means graphically.

b2 – 4ac > 0 2 real solutions 2 x-intercepts

b2 – 4ac = 0 1 real solution Touches the graph, 1 x intercept

b2 – 4ac < 0 2 imaginary solutions No x –intercepts

Find the discriminant. Give the number and type of solutions of the equation.

1. 9x2+ 6x + 1 = 0

2. 9x2+ 6x -4 = 0

3. 9x2+ 6x +5 = 0

Vertical Motion Models

h = -16t2 + ho object is dropped

h = -16t2 + vot + ho object is launched or thrown

Variables:h = height (feet)t = time (seconds)vo = initial velocity

ho = initial height

The water in a large fountain leaves the spout with a vertical velocity of 30 ft per second. After going up in the air, it lands in a basin 6 ft below the spout. If the spout is 10 ft above the ground, how long does it take a single drop of water to travel from the spout to the basin?

A man tosses a penny up into the air above a 100 ft deep well with a velocity of 5 ft/sec. The penny leaves the man’s hand at a height of 4 ft. How long will it take the penny to reach the bottom of the well?