10.6 – Solving quadratic equations with the quadratic formula.
Using the Quadratic Formula Section 3.4 beginning on page 122.
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Transcript of Using the Quadratic Formula Section 3.4 beginning on page 122.
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