Here are my answers. Can you check if I got the right answers? Thank you!
solve for x, where x is a real number.
x^2+3x-4=14
x=6,3
(x^4-1)/(x^3)=0
x=1
(x-5)^2=9
x=2,8
2x^2+5x=8
x= [-5+/-sqrt(89)]/4
for first one x=-6 or x=3, factors are (x+6)(x-3)
second... also x=0
third ... ok
last... ok
about the 2nd. x can't equal to 0, because the denominator can't equal 0
You are right!
Good pickup
Based on your answers, it seems that you have mostly solved the equations correctly. Let me verify each one and explain how you can check your answers.
1. x^2 + 3x - 4 = 14:
You correctly found x = 6 and x = 3 as the solutions. To check your answers, substitute them back into the original equation:
For x = 6:
6^2 + 3(6) - 4 = 14
36 + 18 - 4 = 14
50 - 4 = 14
46 = 14 (This is incorrect)
For x = 3:
3^2 + 3(3) - 4 = 14
9 + 9 - 4 = 14
18 - 4 = 14
14 = 14 (This is correct)
So, the correct solution is x = 3, not x = 6.
2. (x^4 - 1) / (x^3) = 0:
You correctly found x = 1 as the solution. To check your answer, substitute x = 1 into the original equation:
(1^4 - 1) / (1^3) = 0
(1 - 1) / 1 = 0
0 / 1 = 0
0 = 0 (This is correct)
So, x = 1 is indeed the correct solution.
3. (x - 5)^2 = 9:
You correctly found x = 2 and x = 8 as the solutions. To check your answers, substitute them back into the original equation:
For x = 2:
(2 - 5)^2 = 9
(-3)^2 = 9
9 = 9 (This is correct)
For x = 8:
(8 - 5)^2 = 9
(3)^2 = 9
9 = 9 (This is correct)
So, both x = 2 and x = 8 are correct solutions.
4. 2x^2 + 5x = 8:
You correctly found x = (-5 ± √89) / 4 as the solutions. However, you missed the condition that x cannot be equal to 0 due to division by zero. To check your answers, substitute them back into the original equation:
For x = (-5 + √89) / 4:
2((-5 + √89) / 4)^2 + 5((-5 + √89) / 4) = 8
(25 - 10√89 + 89) / 8 + (-25 + 5√89) / 4 = 8
(25 - 10√89 + 89 + 2(-25 + 5√89)) / 8 = 8
(114 - 20√89 - 50 + 10√89) / 8 = 8
(64 - 10) / 8 = 8
54 / 8 = 8
6.75 ≠ 8 (This is incorrect)
For x = (-5 - √89) / 4:
2((-5 - √89) / 4)^2 + 5((-5 - √89) / 4) = 8
(25 + 10√89 + 89) / 8 + (-25 - 5√89) / 4 = 8
(25 + 10√89 + 89 + 2(-25 - 5√89)) / 8 = 8
(114 + 20√89 - 50 - 10√89) / 8 = 8
(64 - 10) / 8 = 8
54 / 8 = 8
6.75 ≠ 8 (This is incorrect)
Based on the checks, neither of the solutions you provided satisfies the original equation. Therefore, x = (-5 ± √89) / 4 is not the correct solution.