x^3-18x^2+80=0
Forgot to mention to solve by factoring!
Thanks in advance
x^3-18x^2+80 factors into
(x+2)(x^2-20x+40)
To solve the equation x^3 - 18x^2 + 80 = 0, you can use a technique called factoring.
Step 1: Look for common factors: In this case, there are no common factors in all the terms.
Step 2: Try to factor out the equation: To do this, we need to find two numbers whose sum is -18 and product is 80. By checking different pairs of numbers that multiply to give 80, we find that -10 and -8 meet these conditions.
Step 3: Rewrite the equation:
x^3 - 10x^2 - 8x^2 + 80 = 0
Step 4: Group the terms:
(x^3 - 10x^2) - (8x^2 - 80) = 0
Step 5: Factor out the greatest common factor from each group:
x^2(x - 10) - 8(x - 10) = 0
Step 6: Notice that both terms have a common binomial factor of (x - 10). Factor it out:
(x^2 - 8)(x - 10) = 0
Step 7: Set each factor equal to zero:
x^2 - 8 = 0 and x - 10 = 0
Step 8: Solve the individual equations:
For the equation x^2 - 8 = 0:
Adding 8 to both sides, we get:
x^2 = 8
Taking the square root of both sides, we get:
x = ±√8
So the solutions are x = √8 and x = -√8, which simplifies to x = ±2√2.
For the equation x - 10 = 0:
Adding 10 to both sides, we get:
x = 10
Step 9: The solutions to the equation x^3 - 18x^2 + 80 = 0 are x = -2√2, x = 2√2, and x = 10.