Find the real or imaginary solutions of the following equation by factoring.
x^3+343=0
We can rewrite the equation as:
(x)^3 + (7)^3 = 0
We can use the sum of cubes formula, which states that: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
So, applying this formula to our equation, we have:
(x + 7)(x^2 - 7x + 7^2) = 0
Now, we can solve for x in each factor:
x + 7 = 0 -> x = -7
x^2 - 7x + 49 = 0
Using the quadratic formula, x = ( -(-7) ± √( (-7)^2 - 4(1)(49) ) ) / (2(1))
Simplifying further, we have x = (7 ± √(-147)) / 2
Since we have a negative value under the square root, the solutions are imaginary.
Thus, the solutions to the equation x^3 + 343 = 0 are -7 and (7 ± √(-147)) / 2.
To find the real or imaginary solutions of the equation x^3 + 343 = 0 by factoring, we'll need to use the difference of cubes formula.
The difference of cubes formula states that for any two numbers a and b, we can factor the expression a^3 - b^3 as (a - b)(a^2 + ab + b^2).
In this case, we have x^3 + 343. We can rewrite 343 as 7^3, so we have the expression x^3 + 7^3.
Using the difference of cubes formula, we can factor x^3 + 343 as (x + 7)(x^2 - 7x + 49).
Now, we can set each factor equal to zero and solve for x:
Setting (x + 7) = 0, we find x = -7.
Setting (x^2 - 7x + 49) = 0, we get a quadratic equation. We can solve this using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
a = 1, b = -7, and c = 49. Substituting these values into the quadratic formula, we have:
x = (7 ± √((-7)^2 - 4(1)(49))) / (2 * 1)
= (7 ± √(49 - 196)) / 2
= (7 ± √(-147)) / 2
Since we have a square root of a negative number, it means that there are no real solutions to the equation x^3 + 343 = 0. However, we do have imaginary solutions.
Therefore, the solutions to the equation x^3 + 343 = 0 are:
- x = -7
- x = (7 ± √(-147)) / 2