Using the Quadratic Formula

Section 3.4 beginning on page 122

The Big IdeasThe quadratic formula allows us to solve any quadratic equation once it is written in standard form. The discriminant is a part of the quadratic formula that tells us how many real solutions the equation has.

Given the quadratic equation

The discriminant

There are two real solutions

There’s exactly one real solution

There are two imaginary solutions

ExamplesExample 4: Find the discriminant of the quadratic equations and describe the number and type of solutions.

a) b) c)

For examples 1-3 first determine how many solutions and the type of solutions, and then find the solutions.

Example 1:

Example 2:

Example 3:

𝑥=−3+√292

≈1.19 𝑥=−3−√292

≈−4.19

𝑥=25

𝑥=2+3 𝑖 𝑥=2−3 𝑖

-4 two imaginary 0 one real 4 two real

Modeling a Launched ObjectThe function is used to model the height of a dropped object. For an object that is launched or thrown, an extra term (in feet per second). Recall that h is the height (in feet(), t is the time in motion (in seconds), and is the initial height (in feet).

Example 6: A juggler tosses a ball into the air. The ball leaves the juggler’s hand 4 feet above the ground and has an initial vertical velocity of 30 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air?

h=−16 𝑡 2+𝑣0 𝑡+h0 𝑣0=¿ h0=¿430 h=33=−16 𝑡 2+30 𝑡+40=−16 𝑡2+30 𝑡+1 𝑡=−𝑏±√𝑏2−4𝑎𝑐

2𝑎

Reject the negative solution because the ball’s time in the air can not be negative. The ball is in the air for about 1.9 seconds.

Solve each equation using the quadratic formula:

1) 2) 3)

4) 5) 6)

Find the discriminant of the quadratic equation and describe the number and type of solutions.

7) 8) 9)

10) 11) 12)

14) A juggler tosses a ball into the air. The ball leaves the juggler’s hand 4 feet above the ground and has an initial vertical velocity of 40 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air?

Monitoring Progress

Answers1) 2) 3)

4) 5) 6)

7) 0; one real 8) 3; two real 9) -196;two imaginary

10) 177; two real 11) -108; two imaginary 12) 0; one real

14) About 2.52 seconds