Solve x^2-x-1=30/x^2-x

$$x ^ { 2 } - x - 1 = \frac { 30 } { x ^ { 2 } - x }$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{19}i+1}{2}\approx 0.5-2.179449472i$$

$$x=\frac{1+\sqrt{19}i}{2}\approx 0.5+2.179449472i$$

$$x=-2$$

$$x=3$$

Solution Steps

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right).

$$x\left(x-1\right)x^{2}-xx\left(x-1\right)+x\left(x-1\right)\left(-1\right)=30$$

Use the distributive property to multiply x by x-1.

$$\left(x^{2}-x\right)x^{2}-xx\left(x-1\right)+x\left(x-1\right)\left(-1\right)=30$$

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

$$x^{4}-x^{3}-xx\left(x-1\right)+x\left(x-1\right)\left(-1\right)=30$$

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

$$x^{4}-x^{3}-x^{2}\left(x-1\right)+x\left(x-1\right)\left(-1\right)=30$$

Use the distributive property to multiply -x^{2} by x-1.

$$x^{4}-x^{3}-x^{3}+x^{2}+x\left(x-1\right)\left(-1\right)=30$$

Combine -x^{3} and -x^{3} to get -2x^{3}.

$$x^{4}-2x^{3}+x^{2}+x\left(x-1\right)\left(-1\right)=30$$

Use the distributive property to multiply x by x-1.

$$x^{4}-2x^{3}+x^{2}+\left(x^{2}-x\right)\left(-1\right)=30$$

Use the distributive property to multiply x^{2}-x by -1.

$$x^{4}-2x^{3}+x^{2}-x^{2}+x=30$$

Combine x^{2} and -x^{2} to get 0.

$$x^{4}-2x^{3}+x=30$$

Subtract 30 from both sides.

$$x^{4}-2x^{3}+x-30=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -30 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±30,±15,±10,±6,±5,±3,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-2x^{3}+x-30 by x+2 to get x^{3}-4x^{2}+8x-15. Solve the equation where the result equals to 0.

$$x^{3}-4x^{2}+8x-15=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -15 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±15,±5,±3,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-4x^{2}+8x-15 by x-3 to get x^{2}-x+5. Solve the equation where the result equals to 0.

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

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 1 for a, -1 for b, and 5 for c in the quadratic formula.

$$x=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\times 5}}{2}$$

Do the calculations.

$$x=\frac{1±\sqrt{-19}}{2}$$

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

$$x=\frac{-\sqrt{19}i+1}{2} x=\frac{1+\sqrt{19}i}{2}$$

List all found solutions.

$$x=-2 x=3 x=\frac{-\sqrt{19}i+1}{2} x=\frac{1+\sqrt{19}i}{2}$$

Solve for x

$$x=3$$

$$x=-2$$

Solution Steps

Variable x cannot be equal to any of the values 0,1 since division by zero is not defined. Multiply both sides of the equation by x\left(x-1\right).

$$x\left(x-1\right)x^{2}-xx\left(x-1\right)+x\left(x-1\right)\left(-1\right)=30$$

Use the distributive property to multiply x by x-1.

$$\left(x^{2}-x\right)x^{2}-xx\left(x-1\right)+x\left(x-1\right)\left(-1\right)=30$$

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

$$x^{4}-x^{3}-xx\left(x-1\right)+x\left(x-1\right)\left(-1\right)=30$$

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

$$x^{4}-x^{3}-x^{2}\left(x-1\right)+x\left(x-1\right)\left(-1\right)=30$$

Use the distributive property to multiply -x^{2} by x-1.

$$x^{4}-x^{3}-x^{3}+x^{2}+x\left(x-1\right)\left(-1\right)=30$$

Combine -x^{3} and -x^{3} to get -2x^{3}.

$$x^{4}-2x^{3}+x^{2}+x\left(x-1\right)\left(-1\right)=30$$

Use the distributive property to multiply x by x-1.

$$x^{4}-2x^{3}+x^{2}+\left(x^{2}-x\right)\left(-1\right)=30$$

Use the distributive property to multiply x^{2}-x by -1.

$$x^{4}-2x^{3}+x^{2}-x^{2}+x=30$$

Combine x^{2} and -x^{2} to get 0.

$$x^{4}-2x^{3}+x=30$$

Subtract 30 from both sides.

$$x^{4}-2x^{3}+x-30=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -30 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±30,±15,±10,±6,±5,±3,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-2x^{3}+x-30 by x+2 to get x^{3}-4x^{2}+8x-15. Solve the equation where the result equals to 0.

$$x^{3}-4x^{2}+8x-15=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -15 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±15,±5,±3,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-4x^{2}+8x-15 by x-3 to get x^{2}-x+5. Solve the equation where the result equals to 0.

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

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 1 for a, -1 for b, and 5 for c in the quadratic formula.

$$x=\frac{-\left(-1\right)±\sqrt{\left(-1\right)^{2}-4\times 1\times 5}}{2}$$

Do the calculations.

$$x=\frac{1±\sqrt{-19}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$x\in \emptyset$$

List all found solutions.

$$x=-2 x=3$$