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x^{2}x+x^{2}\left(-3\right)+4=0
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}.
x^{3}+x^{2}\left(-3\right)+4=0
To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.
x^{3}-3x^{2}+4=0
Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.
±4,±2,±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
x=-1
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x^{2}-4x+4=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-3x^{2}+4 by x+1 to get x^{2}-4x+4. Solve the equation where the result equals to 0.
x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 1\times 4}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -4 for b, and 4 for c in the quadratic formula.
x=\frac{4±0}{2}
Do the calculations.
x=2
Solutions are the same.
x=-1 x=2
List all found solutions.