what are the real solutions of
750=x^3-15x^2+50x
A theorem call the "rational root" or "p/q" theorem will tell you that, if there is a real root, it must be p/q, where p and q are even divisors of the constant term (750) and highest-order term (1).
http://en.wikipedia.org/wiki/Rational_root_theorem
Using this, you will find that 15 is one root and the other factor (besides x-15) is x^2 + 50.
x^2 + 50 = 0 has no real roots.
Therefore there is only one solution.
To find the real solutions of the equation 750 = x^3 - 15x^2 + 50x, we need to solve the equation for x. Here's how you can do it:
Step 1: Rearrange the equation to make it equal to zero:
x^3 - 15x^2 + 50x - 750 = 0
Step 2: Try to factor the equation. Look for any common factors among the terms. In this case, there are no obvious common factors, so we move on to the next step.
Step 3: Use numerical methods to solve the equation. One popular method is the Newton-Raphson method, which is an iterative process that can give us accurate solutions.
However, since the equation is a cubic equation, there is a simpler method that can be used. This is called the Rational Root Theorem.
Step 4: Apply the Rational Root Theorem. The Rational Root Theorem states that if there is a rational solution to the polynomial equation with integer coefficients, it will be in the form of p/q, where p is a factor of the constant term (750 in this case), and q is a factor of the leading coefficient (1 in this case).
For the equation x^3 - 15x^2 + 50x - 750 = 0, the factors of 750 are ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±25, ±30, ±50, ±75, ±125, ±150, ±250, ±375, and ±750. The factors of 1 are ±1.
Step 5: Substitute each of the possible rational roots into the equation to check if any of them satisfy the equation. To do this, you can use synthetic division or simply substitute the value into the equation and check if it equals zero.
After checking each possible root, the real solutions that satisfy the equation can be identified.