Answer:
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:
- 0 = 0 [tex]\checkmark[/tex]
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 [tex]\checkmark[/tex]
ii) x = 3/4 ⇒ 4(3/4)² - 3(3/4) =3²/4 - 9/4 = 9/4 - 9/4 = 0 [tex]\checkmark[/tex]
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 [tex]\checkmark[/tex]
ii) x = - 3 ⇒ (-3)² + 5(-3) + 6 = 9 - 15 + 6 = 9 - 9 = 0 [tex]\checkmark[/tex]
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 [tex]\checkmark[/tex]
ii) n = 1 ⇒ (1)² - 5(1) + 4 = 1 - 5 + 4 = 1 - 1 = 0[tex]\checkmark[/tex]
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 [tex]\checkmark[/tex]
ii) x = 2/3 ⇒ 3(2/3)² + (2/3) - 2 = 4/3 + 2/3 - 2 = 6/3 - 2 = 0 [tex]\checkmark[/tex]
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 [tex]\checkmark[/tex]
ii) x = 1/2 ⇒ 6(1/2)² + 3(1/2) - 3 = 6/4 + 3/2 - 3 = 3 - 3 = 0 [tex]\checkmark[/tex]
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:
- Area = (x - 4)(x + 1) = 66
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:
- (10 - 4)(10 + 1) = 6 × 11 = 66 cm² [tex]\checkmark[/tex]
