how do you solve 2x^2+x^3=96?
see
http://www.jiskha.com/display.cgi?id=1319501043
To solve the equation 2x^2 + x^3 = 96, we can follow these steps:
Step 1: Rearrange the equation in standard form: x^3 + 2x^2 - 96 = 0
Step 2: Factor out any common factors, if possible. In this case, the equation does not have any common factors.
Step 3: Check for any special factoring patterns. However, there are no special factoring patterns applicable to this equation.
Step 4: Use the Rational Root Theorem to find possible rational roots. The Rational Root Theorem states that if the polynomial has any rational roots, they will be of the form p/q, where p is a factor of the constant term (-96) and q is a factor of the leading coefficient (1). Possible rational roots are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±32, ±48, ±96.
Step 5: Test the possible rational roots by substituting them into the equation. The goal is to find any roots that satisfy the equation.
Let's try the first possible root, x = 1:
(1)^3 + 2(1)^2 - 96 = 1 + 2 - 96 = -93
Since -93 is not equal to zero, x = 1 is not a root.
Now, let's try the next possible root, x = -1:
(-1)^3 + 2(-1)^2 - 96 = -1 + 2 - 96 = -95
Again, -95 is not equal to zero, so x = -1 is not a root.
We repeat this process for all the possible rational roots until we find one that satisfies the equation. In this case, you will find that there are no rational roots that satisfy the equation.
Step 6: Since there are no rational roots, we can use numerical methods to approximate the solutions. One common numerical method is to use a graphing calculator or software to plot the function and find the x-values where it intersects the x-axis, which correspond to the solutions of the equation.
Alternatively, we can also use numerical methods like polynomial approximation or iterative methods to approximate the solutions.
In conclusion, solving the equation 2x^2 + x^3 = 96 requires using numerical methods to approximate the solutions since there are no rational roots.