Solve x^2=13/9+2/3x

$${ x }^{ 2 } = \frac{ 13 }{ 9 } + \frac{ 2 }{ 3x }$$

Solve for x

$$x=\frac{1-\sqrt{10}}{3}\approx -0.72075922$$

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x = \frac{\sqrt{10} + 1}{3} \approx 1.387425887$$

Solution Steps

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 9x, the least common multiple of 9,3x.

$$9xx^{2}=9x\times \frac{13}{9}+3\times 2$$

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

$$9x^{3}=9x\times \frac{13}{9}+3\times 2$$

Multiply 9 and \frac{13}{9} to get 13.

$$9x^{3}=13x+3\times 2$$

Multiply 3 and 2 to get 6.

$$9x^{3}=13x+6$$

Subtract 13x from both sides.

$$9x^{3}-13x=6$$

Subtract 6 from both sides.

$$9x^{3}-13x-6=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -6 and q divides the leading coefficient 9. List all candidates \frac{p}{q}.

$$±\frac{2}{3},±2,±6,±\frac{1}{3},±1,±3,±\frac{2}{9},±\frac{1}{9}$$

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.

$$x=-\frac{2}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 9x^{3}-13x-6 by 3\left(x+\frac{2}{3}\right)=3x+2 to get 3x^{2}-2x-3. Solve the equation where the result equals to 0.

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

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 3 for a, -2 for b, and -3 for c in the quadratic formula.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 3\left(-3\right)}}{2\times 3}$$

Do the calculations.

$$x=\frac{2±2\sqrt{10}}{6}$$

Solve the equation 3x^{2}-2x-3=0 when ± is plus and when ± is minus.

$$x=\frac{1-\sqrt{10}}{3} x=\frac{\sqrt{10}+1}{3}$$

List all found solutions.

$$x=-\frac{2}{3} x=\frac{1-\sqrt{10}}{3} x=\frac{\sqrt{10}+1}{3}$$