These are 3 questions and 3 answers:
Question 1) p^2 - 2p - 15 = 0
Solution:
1) factoring:
a) write two parenthesis with p as the first term (common term) of each:
(p ) (p )
b) in the first parenthesis copy the sign of the second term (i.e. - ), and in the second parenthesis put the product of the signs of the second and third terms (i.e. - * - = +)
=> (p - )(p + )
c) search two numbers whose sum is - 2 and its product is - 15. Those numbers are -5 and +3, look:
- 5 + 3 = - 2
( - 5)( + 3) = - 15
=> (p - 5)(p + 3)
d) you can prove that (p - 5) (p + 3) = p^2 - 20 - 15
2) equal the expression to zero and solve:
(p - 5)(p + 3) = 0 => (p - 5) = 0 or (p + 3) = 0
p - 5 = 0 => p = 5
p + 3 = 0 => p = - 3
=> that means that both p = 5 and p = - 3 are solutions.
3) check the solutions:
p = 5 => (5)^2 - 2(5) - 15 = 25 - 10 - 15 = 0 => check
p = - 3 => (-3)^2 - 2(-3) - 15 = 9 + 6 - 15 = 0 => check
4) As a solution set: {5, - 3}
Question 2: (x - 1)(x + 2) = 18
Solution
0) expand, combine like terms and transpose 18:
x^2 + x - 2 - 18 = 0
x^2 + x - 20 = 0
1) factoring:
a) write two parenthesis with x as the first term (common term) of each:
(x ) (x )
b)
in the first parenthesis copy the sign of the second term (i.e. + ),
and in the second parenthesis put the product of the signs of the second
and third terms (i.e. + * - = -)
=> (x + )(x - )
c) search two numbers whose sum is + 1 and its product is - 20. Those numbers are + 5 and - 4, look:
5 - 4 = 1
(5)( - 4) = - 20
=> (x + 5)(x - 4)
d) you can prove that (x + 5) (x - 4)) = x^2 + x - 20
2) equal the expression to zero and solve:
(x + 5)(x - 4) = 0 => (x + 5) = 0 or (x - 4) = 0
x + 5 = 0 => x = - 5
x - 4 = 0 => x = 4
=> that means that both x = - 5 and x = 4 are the solutions.
3) check the solutions:
x = - 5 => (- 5 - 1)(- 5 + 2) = 18
=> (-6)(-3) = 18
=> 18 = 18 => check
x = 4 => (4 - 1)(4 + 2) = (3)(6) = 18 => check
4) As a solution set: {- 5, 4}
Question 3: (k - 6) (k - 1) = - k - 2
Solution:
0) (k - 6)(k - 1) = - k - 2 =>
k^2 -7k + 6 = - k - 2 =>
k^2 -7x + k + 6 + 2 = 0 =>
k^2 - 6k + 8 = 0
1) factor
(k - ) (k - ) = 0
(k - 4) (k - 2) = 0 <-------- - 4 - 2 = - 6 and (-4)(-2) = +8
2) Zero product property:
k - 4 = 0 , k - 2 = 0
=> k = 4, k = 2
3) Check:
a) (4 - 6) (4 - 1) = - 4 - 2
(-2)(3) = - 6
- 6 = - 6 => check
b) (2 - 6)(2 - 1) = - 2 - 2
(-4)(1) = -4
-4 = - 4 => check
4) Solution set: {2, 4}
