Solve x^2=21-24-x/x

$${ x }^{ 2 } =21- \frac{ 24-x }{ x }$$

Solve for x

$$x=\sqrt{10}-2\approx 1.16227766$$

$$x=-\sqrt{10}-2\approx -5.16227766$$

$$x=4$$

Solution Steps

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

$$xx^{2}=x\times 21-\left(24-x\right)$$

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

$$x^{3}=x\times 21-\left(24-x\right)$$

To find the opposite of 24-x, find the opposite of each term.

$$x^{3}=x\times 21-24+x$$

Combine x\times 21 and x to get 22x.

$$x^{3}=22x-24$$

Subtract 22x from both sides.

$$x^{3}-22x=-24$$

Add 24 to both sides.

$$x^{3}-22x+24=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 24 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±24,±12,±8,±6,±4,±3,±2,±1$$

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.

$$x=4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-22x+24 by x-4 to get x^{2}+4x-6. Solve the equation where the result equals to 0.

$$x^{2}+4x-6=0$$

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 -6 for c in the quadratic formula.

$$x=\frac{-4±\sqrt{4^{2}-4\times 1\left(-6\right)}}{2}$$

Do the calculations.

$$x=\frac{-4±2\sqrt{10}}{2}$$

Solve the equation x^{2}+4x-6=0 when ± is plus and when ± is minus.

$$x=-\sqrt{10}-2 x=\sqrt{10}-2$$

List all found solutions.

$$x=4 x=-\sqrt{10}-2 x=\sqrt{10}-2$$