Completing the Square. The Quadratic Formula and the Discriminant.

What you’ll learn

To solve quadratic equations by completing the square. To solve quadratic equations using the quadratic formula. To find the number of solutions of a quadratic equation.

Vocabulary

Completing the square. Quadratic formula.Discriminant.

Take a note: A very important note:

In general, you can change the expression into a perfect square trinomial by adding This process is called completing the square.The process is the same whether b is positive or negative.

bxx 2 bxx to

2

b 22

Example: cx16x 2

642

16

2

22

b

64c so

Problem 1: Solving

cbxx 2 What are the solutions of the equation ?216x6x 2

216x6x 2 92169x6x 2

2253x 2

2253x 2

153x -153x or 153x

-18x 12x

side each to 9 2

6 Add

2

squarea as 96x xWrite 2

Simplify

sides both to rootsquare Find

equations two asWrite

Solve

Your turn 247t6t 2 Answer -13,19

Problem 2: Solving

0cbxx 2 What are the solutions of the equation ?016x14x 2

16x14x 2

491649x14x 2 337x 2

337x 74.57x

-5.747-x or 74.57x

1.26x 12.74 x

7-5.74x 75.74x

492

14- Add

2

squarea as 49x14x Write 2

33 calculatora Use

equations two as Write

Solve

side each to rootsquare the Find

Your turn015x9x 2 Answer -2.21,-6.79

Take a note: You can find the solution(s) of any quadratic equation using the quadratic formula

then 0,a and,0cbxax if 2

Example: 05x3 2xSuppose 2

𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎

𝑥=−3±√32−4 (2)(−5)

2(2)¿ −3±√ 49

4

𝑥=−3+74

¿−3±74

𝑥=−3−74

¿−104¿

44

𝑥=1 𝑥=−52

Problem 3: Using the Quadratic Formula

What are the solutions of .Use the QF

x28x 2

x28x 2

𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎

08x2x 2 Make the equation =0

𝑥=−(−2)±√(−2)2−4 (1)(−8)

2(1)¿ 4 ±√36

2

42

62x

2

2

6-2x

Your turn

010x3x)b

21x4x)a2

2

What are the solutions of .Use the QF

Answers: a)-3,7b) -2,5

Your turn again:Which method would you choose to solve each equation?

0273)

0154)

0107)

011173)

0322)

2

2

2

2

2

xe

xxd

xxc

xxb

xaAnswers

rootssquare )16x(2)a 2

QF)b

5)2)(x(x Factoring)c squarethe completing or QF)d

rootsquare )9x(3)e 2

In the shot put, an athlete throws a heavy metal ball through the air. The arc of the ball can be model by the equation where x is the horizontal distance, in meters, from the athlete and y is the height, in meters of the ball. Howfar from the athlete will the ball land?

Problem 4: Finding Approximate Solutions.

𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎

𝑥=−0.84 ±√0.842−4 (−0.04 )(2)

2(−0.04 )16.23x

16.2x

2x84.0x044.0y 2

A batter strikes a baseball.The equation models its path, where x is the horizontal distance in ft., the ball travels and y is the height, in feet, of the ball. How far from the batter will the ball land?Rounded to nearest tenth of a foot.

Your turn

Answer:

144.8 ft

5.3x7.0x005.0y 2

Take a note: In this formula 𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎

nt.discrimina called is ac4b2

Problem 5: Using the Discriminant

How many real solutions does 5x3x2 2

0 equationthe Make 05x3x2 2

ac4b2 315243 2

Because the discriminant is negative has no real solutions

Your turn7x5x6 2 Answer: 2

Classwork odd Homework evenTB pgs. 564-565 exercises 7-30 and 34-42 pgs. 571-572 exercises 7-40