Solve 5x^2+2/x=10+x

$$5 x ^ { 2 } + \frac { 2 } { x } = 10 + x$$

Solve for x

$$x=\sqrt{2}\approx 1.414213562$$

$$x=-\sqrt{2}\approx -1.414213562$$

$$x=\frac{1}{5}=0.2$$

Solution Steps

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

$$5x^{2}x+2=x\times 10+xx$$

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

$$5x^{3}+2=x\times 10+xx$$

Multiply x and x to get x^{2}.

$$5x^{3}+2=x\times 10+x^{2}$$

Subtract x\times 10 from both sides.

$$5x^{3}+2-x\times 10=x^{2}$$

Subtract x^{2} from both sides.

$$5x^{3}+2-x\times 10-x^{2}=0$$

Multiply -1 and 10 to get -10.

$$5x^{3}+2-10x-x^{2}=0$$

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

$$5x^{3}-x^{2}-10x+2=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 2 and q divides the leading coefficient 5. List all candidates \frac{p}{q}.

$$±\frac{2}{5},±2,±\frac{1}{5},±1$$

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.

$$x=\frac{1}{5}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 5x^{3}-x^{2}-10x+2 by 5\left(x-\frac{1}{5}\right)=5x-1 to get x^{2}-2. Solve the equation where the result equals to 0.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\times 1\left(-2\right)}}{2}$$

Do the calculations.

$$x=\frac{0±2\sqrt{2}}{2}$$

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

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

List all found solutions.

$$x=\frac{1}{5} x=-\sqrt{2} x=\sqrt{2}$$