3.5 Quadratic Equations

OBJ: To solve a quadratic equation by factoring

DEF: Standard form of a quadratic equation

ax2 + bx + c = 0

• NOTE: Each equation contains a

polynomial of the second degree.

DEF: Zero – product property

If mn = 0, then m = 0 or n = 0 or both = 0

• NOTE: Solve some quadratic equations by:• Writing equation in standard form

• Factoring

• Setting each factor equal to 0

EX: 5 c 2 + 7c – 6 = 0

5 3 5 2

6 2 1 3

-3

5 3

6 2

+10

(5c – 3)(c + 2) = 0

c = 3/5, -2

EX: 7t = 20 – 3 t 2

3 t 2 + 7t – 20 = 03 4 3 54 5 1 4 -53 54 4 +12 (3t – 5)(t + 4) = 0t = 5/3, -4

EX: 36 = 25 x 2

25 x 2 – 36 = 0

(5x – 6)(5x + 6) = 0

x = ± 6/5

EX: –2 x 2 = 5x

2 x 2 + 5x = 0

x(2x + 5) = 0

x = 0, - 5/2

EX: 7 n 2 + 14n – 56 = 0

7 (n 2 + 2n – 8) = 0

7 (n + 4)(n – 2) = 0

n = - 4, 2

EX: y 4 – 5 y 2 + 4 = 0

(y 2 – 4)(y 2 – 1) = 0

(y – 2)(y + 2)(y – 1)(y + 1) = 0

Y = ± 2, ± 1

EX: y 4 – 10 y 2 + 9 = 0

(y 2 – 9)(y 2 – 1) = 0

(y – 3)(y + 3)(y – 1)(y + 1) = 0

Y = ± 3, ± 1

EX: y 4 = 20 – y 2

y 4 + y 2 – 20 = 0

(y 2 + 5)(y 2 – 4) = 0

(y 2 + 5)(y – 2)(y + 2) = 0

Y = ± i√ 5, ± 2

EX: y 4 = 12 + y 2

y 4 – y 2 – 12 = 0

(y 2 – 4)(y 2 + 3) = 0

(y – 2)(y + 2)(y 2 + 3) = 0

Y = ± 2, ± i√ 3

6.1 Square Roots

OBJ: To solve a quadratic equation by using the definition of square root

DEF: Square root

If x 2 = k, then x = ±√k, for k ≥ 0

EX: 6 y 2 – 20 = 8 – y 2

7y 2 = 28

y 2 = 4

y = ± 2

EX: 3 n 2 + 9 = 7 n 2 – 35

44 = 4n 2

11 = n 2

±√11 = n

7.3 The Quadratic Formula

OBJ: To solve a quadratic equation by using the quadratic formula

DEF: The quadratic formula

x = -b ± √b2 – 4ac

2a

EX: 4 x 2 – 7x + 2 = 0

x = -(-7) ± √(-7)2 – 4(4)(2)

2(4)

= 7 ± √49 – 32

8

= 7 ± √17

8

EX: 9 x 2 = 12x – 1

9 x 2 – 12x + 1 = 0x = -(-12)±√(-12)2 – 4(9)(1)

2(9) = 12 ± √144 – 36 18 = 12 ± √108 18 = 12 ± 6√3 18

= 12 ± 6√3

18

= 6(2 ± √3)

18

3

= 2 ± √3

3

EX: 6 x 2 + 5x = 0

x(6x + 5) = 0

x = 0, -5/6

EX 8: 72 – x 2 = 0

x 2 = 72

x = ± 6√2

8.3 Equations With Imaginary Number Solutions

OBJ: To solve an equation whose solutions are imaginary

EX: 2 x 2 + 7 = 6x

2 x 2 – 6x + 7 = 0x = -(-6)±√(-6)2 – 4(2)(7) 2(2) = 6 ± √36 – 56 4 = 6 ± √-20 4 = 6 ± 2i√5 4

= 2(3 ± i√5) 4 = 2(3 ± i√5) 4 2 = 3 ± i√5 2

EX: 27 – 6 y 2 = y 4

y 4 + 6 y 2 – 27 = 0

(y 2 + 9)(y 2 – 3) = 0

y = ± 3i, ± √3