Solve x^2+5*x^2/x-2left(x-2right)=5

$${ x }^{ 2 } +5 \times \frac{ { x }^{ 2 } }{ x-2 } \left( x-2 \right) = 5$$

Solve for x

$$x=\frac{\sqrt{30}}{6}\approx 0.912870929$$

$$x=-\frac{\sqrt{30}}{6}\approx -0.912870929$$

Solution Steps

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

$$\left(x-2\right)x^{2}+5x^{2}\left(x-2\right)=5\left(x-2\right)$$

Use the distributive property to multiply x-2 by x^{2}.

$$x^{3}-2x^{2}+5x^{2}\left(x-2\right)=5\left(x-2\right)$$

Use the distributive property to multiply 5x^{2} by x-2.

$$x^{3}-2x^{2}+5x^{3}-10x^{2}=5\left(x-2\right)$$

Combine x^{3} and 5x^{3} to get 6x^{3}.

$$6x^{3}-2x^{2}-10x^{2}=5\left(x-2\right)$$

Combine -2x^{2} and -10x^{2} to get -12x^{2}.

$$6x^{3}-12x^{2}=5\left(x-2\right)$$

Use the distributive property to multiply 5 by x-2.

$$6x^{3}-12x^{2}=5x-10$$

Subtract 5x from both sides.

$$6x^{3}-12x^{2}-5x=-10$$

Add 10 to both sides.

$$6x^{3}-12x^{2}-5x+10=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 10 and q divides the leading coefficient 6. List all candidates \frac{p}{q}.

$$±\frac{5}{3},±\frac{10}{3},±5,±10,±\frac{5}{6},±\frac{5}{2},±\frac{1}{3},±\frac{2}{3},±1,±2,±\frac{1}{6},±\frac{1}{2}$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 6x^{3}-12x^{2}-5x+10 by x-2 to get 6x^{2}-5. Solve the equation where the result equals to 0.

$$6x^{2}-5=0$$

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 6 for a, 0 for b, and -5 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 6\left(-5\right)}}{2\times 6}$$

Do the calculations.

$$x=\frac{0±2\sqrt{30}}{12}$$

Solve the equation 6x^{2}-5=0 when ± is plus and when ± is minus.

$$x=-\frac{\sqrt{30}}{6} x=\frac{\sqrt{30}}{6}$$

Remove the values that the variable cannot be equal to.

$$x\in \emptyset$$

List all found solutions.

$$x=2 x=-\frac{\sqrt{30}}{6} x=\frac{\sqrt{30}}{6}$$

Variable x cannot be equal to 2.

$$x=\frac{\sqrt{30}}{6} x=-\frac{\sqrt{30}}{6}$$