ACTIVITY 12Quadratic Equations (Section 1.3, pp. 97-105)

Quadratics and Intergal DomainsA quadratic equation is an equation of the form:

ax2 + bx + c = 0where a, b, and c are real numbers with a ≠ 0.

Zero-Product Property: For any A,B ∈ R:

AB = 0 if and only if A = 0 or B = 0.

Example 1:

Solve the following equations by factoring:x2 − 7x + 12 = 0

043 xx So by the zero-product property we have 03 x or (but in our case

and) 04 x

That is that x = 3 and x = 4

32 2 zz032 2 zz

03322 2 zzz 01312 zzz

0132 zz 032 z 01 zand

1z32 z

2

3z

Completing the Square:

To make a perfect square out of x2 + bx,

add the square of half the coefficient of x, that is

(b/2)2. Thus:

222

22x

bx

bbx

Example 2:

Solve each equation by completing the square:

0 6 -4x x2 We need (4/2)2=4 but we have a -6

0 6 -4-44x x2 0 6 -4-44x x2

0 6 -4-2 2 x

0 10-2 2 x

102 2 x

102 x

102 x

0 1 -6x - 3x2 0 1 -2x - x3 2

0 1 - 1-12x - x3 2 0 1 -3- 12x - x3 2

0 1 -3- 1-3 2 x

0 4- 1-3 2 x

4 1-3 2 x

3

4 1- 2 x

3

4 1- x

3

41 x

Example 3:

Find all solutions of each equation:

0 4 7x 3x2

0 4 x3

7 x3 2

0 4 36

49

36

49x

3

7 x3 2

0 4 12

49

36

49x

3

7 x3 2

0 4 12

49

36

49x

3

7 x3 2

0 12

12*4

12

49

6

7 x3

2

0 12

48

12

49

6

7 x3

2

12

1

6

7 x3

2

3121

6

7 x

2

3

1*

12

1

6

7 x

2

36

1

6

7 x

2

36

1

6

7 x

6

1

6

7 x

6

17-

6

17- x

6

6- 1

and6

17- x

6

8

3

4

6

1

A closer look at roots

For the polynomial 3x2+7x+4 we saw that x=-1 and x=-4/3 are roots. Does this help us factor this

polynomial?

Let look as a simpler example: suppose that we have x2-1

11 xxSince this is the difference of two squares we know that it factors to

So if we wish to find the roots we need to set our polynomial equal to zero 011 xx

And solve 01 x 01 x

1x 1x

Consequently

If c is a root then (x-c) is a factor and conversely. For the polynomial 3x2+7x+4 we saw that x=-1 and x=-4/3 are roots.

So (x-(-1)) = x+1 and (x-(-4/3)) = x+(4/3) are both factors

3

413473 2 xxxx

3

4

3

42 xxx3

4

3

72 xx

THIS IS VERY CLOSE TO 3x2+7x+4 WE JUST NEED TO MULTIPLY EVERYTHING BY 3!

Consequently,

431 xx

3

41 xx

0 3 5t 2t2

0 3 t2

5 t2 2

0 3 16

25

16

25t

2

5 t2 2

0 3 16

2*25

16

25t

2

5 t2 2

0 3 8

25

4

5 t 2

2

0 8

24

8

25

4

5 t 2

2

0 8

1

4

5 t 2

2

8

1

4

5 t 2

2

16

1

4

5 t

2

16

1

4

5 t

2

16

1

4

5 t

4

1

4

5 t

4

15- t

4

15- t

1t

4

15- tand

4

6-t

2

3-

352 2 tt

2

312 tt

321 tt

xx

41

LCD = x

42 xx

042 xx

The Discriminant:

The discriminant D of the quadratic equation

ax2 + bx + c = 0, where a ≠ 0, is:

D = b2 − 4ac1. If D > 0 the eq. has 2 distinct real roots.2. If D = 0 the eq. has exactly 1 real root.3. If D < 0 the eq. has no real roots.

Example 4:

Use the discriminant to determine how manyreal roots each equation has. Do not solve the equation.

0 1 5x - 3x2

A= 3

B= -5

C= 1

1*3*45- D 2 1252 D 0

Thus, there are TWO real solutions!

x2 = 6x − 1001062 xx

A= 1

B= -6

C= 10

10*1*46- D 2 4063 D

Thus, there are NO real solutions!

0

Example 5:

Find all values of k that ensure that the equationkx2 + 36x + k = 0

has exactly one root (solution).

A= k

B= 36C= k

kk **436 D 2 0041296 2 k

241296 k2324 k

324k18k

Example 7 (Falling-Body Problem): An object is thrown straight upward at an

initial speed of 400 ft/s. From Physics, it is known that, after t seconds, it reaches a height of h feet given by the formula:

h = −16t2 + 400t.When does the object fall back to ground level?

400t 16t- 0 2 25)-16t(t - 0

16t- 0 25)-(t 0

t0 25t

(b) When does it reach a height of 1,600 ft?

400t 16t- 6001 2

1600-400t 16t- 0 2 10025t t0 2

520 0 tt

5t 20t

(d) How high is the highest point the object reaches?

12.5400 12.516- h 2 5000 2500- h 5002 Feet