3. Resuelve las siguientes ecuaciones cuadráticas.

4x2=0
4x2-3x=0
x2+5x+6=0
n2-5n+4=0
3x2+x-2=0
6x2+3x-3=0

4. Dado el siguiente rectángulo, determina cuál debe ser el valor de x que permita obtener el área que se indica.

Explanation:

1. 4x²=0

a) Divide both sides by 4:

b) Take square root of both sides

c) Solution: there is only one solution: x = 0.

d) Substitute to verify:

2. 4x² - 3x = 0

a) Common factor x:

b) Zero product property: any or both factors must equal zero

i) x = 0

ii) 4x - 3 = 0

   4x = 3

     x = 3/4

Solutions x = 0 and x = 3

c) Verify the solutions:

i) x = 0 ⇒ 4(0)² - 3(0) = 0 - 0 = 0

ii) x = 3/4 ⇒ 4(3/4)² - 3(3/4) =3²/4 - 9/4 = 9/4 - 9/4 = 0

3. x² + 5x + 6 = 0

a) Factor: find two numbers whose sum is 5 and product is 6: 3 and 2:

b) Zero product property:

i) x  + 2 = 0

  x = - 2

ii) x + 3 = 0

   x = - 3

The two solutions are x = - 2 and x = -3

c) Verify

i) x = - 2 ⇒ (-2)² + 5(-2) + 6 = 4 - 10 + 6 = - 6 + 6 = 0

ii) x = - 3 ⇒ (-3)² + 5(-3) + 6 = 9 - 15 + 6 = 9 - 9 = 0

4. n² - 5n + 4 = 0

a) Factor: find two numbers whose sum is -5 and product is 4: -4 and -1

b) Zero product property:

i) n - 4 = 0 ⇒ n = 4

ii) n - 1 = 0 ⇒ n = 1

The two solutions are n = 4 and n = 1

c) Verify the solutions:

i) n = 4 ⇒(4)² - 5(4) + 4 = 16 - 20 + 4 = 16 - 16 = 0

ii) n = 1 ⇒ (1)² - 5(1) + 4 = 1 - 5 + 4 = 1 - 1 = 0

5. 3x² + x - 2 = 0

a) Substitute x with 3x - 2x

b) Group the terms (associative property)

c) Factor each group:

d) Common factor x + 1:

e) Zero product property

i) x + 1 = 0 ⇒ x = - 1

ii) 3x - 2 = 0 ⇒ x = 2/3

The two solutions are x = -1 and x = 2/3

f) Verify the solutions:

i) x = - 1 ⇒ 3(-1)² + (-1) - 2 = 3 - 1 - 2 = 2 - 2 = 0

ii) x = 2/3 ⇒ 3(2/3)² + (2/3) - 2 = 4/3 + 2/3 - 2 = 6/3 - 2 = 0

6. 6x² + 3x - 3 = 0

a) Divide both side by 3:

b) Substiture x with 2x - x

c) Group terms:

d) Factor each binomial

e) Common factor x + 1

f) Zero product property

i) x + 1 = 0 ⇒ x = - 1

ii) 2x + 1 = 0

     x = 1/2

The two solutions are x = - 1 and x =  1/2

e) Verify

i) x = - 1 ⇒ 6(-1)² + 3(-1) - 3 = 6 - 3 - 3 = 0

ii) x = 1/2 ⇒ 6(1/2)² + 3(1/2) - 3 = 6/4 + 3/2 - 3 = 3 - 3 = 0

7. Problem

4. Dado el siguiente rectángulo, determina cuál debe ser el valor de x que permita obtener el área que se indica.

The area of a rectangle is the product of the length and the width:

Solve the equation:

a) Distributive property:

b) Add like terms:

c) Subtract 66 from both sides

d) Factor: find two numbers whose sum is -3 and product is - 70: - 10 and 7

e) Zero product property:

i) x - 10 = 0 ⇒ x = 10

ii) x + 7 = 0 ⇒ x = -7

x = - 7 is not a valid solution because that would make the side lengths negative.

Thus, the only solution is x = 10 cm

f) Verify:

the answe i =−n+5

b

step-by-step explanation: