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In answer to your first question, anything with an equal sign is an equation if the values on either side are actually equal. To assume that they are equal and solve it is the foundation of math. If the two values were not equal, it would not be an equation and we could not solve it.
Of course this is an equation. Only it isn't a linear or a quadratic one, instead it is a cubic equation.
x^3 = 2197
Now we need values of x for which x^3 = 2197
We see that 2197 can be written as 13^3
x^3 = 13^3
As the exponent is the same, we can equate the base
=> x = 13
Also, as this is a cubic equation and (-1)^3 = -1, there are no other roots.
So, you are looking for the cube root of 2197, right? The cube root of 2197 is 13. You can test this out simply by finding what the cube of 13 is.
13 x 13 = 169
169 x 13 = 2197
So, 13 is the cube root of 2197.
Yes, it is!
We notice that the number is the cube of 13:
2197 = 13^3.
We'll re-write the equation as a difference of squares:
x^3 - 13^3 = 0
We'll apply the formula:
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
x^3 - 13^3 = (x-13)(x^2 + 13x + 169)
If x^3 - 13^3 = 0 => (x-13)(x^2 + 13x + 169) = 0
We'll set each factor as zero:
x -13= 0
x = 13
x^2 + 13x + 169 = 0
x1 = [-12+sqrt(169 - 676)]/2
Since sqrt-507 is not a real value, the equation has a single real solution.
The real solution of the equation is x = 13
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