Solve x-3+4/x^2=0

$$x-3+ \frac{ 4 }{ { x }^{ 2 } } =0$$

Solve for x

$$x=2$$

$$x=-1$$

Solution Steps

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}.

$$x^{2}x+x^{2}\left(-3\right)+4=0$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{3}+x^{2}\left(-3\right)+4=0$$

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{3}-3x^{2}+4=0$$

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}.

$$±4,±2,±1$$

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.

$$x=-1$$

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.

$$x^{2}-4x+4=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$x=\frac{4±0}{2}$$

Solutions are the same.

$$x=2$$

List all found solutions.

$$x=-1 x=2$$