Solve x^2-36=x^2(x-36x)

$$x ^ { 2 } - 36 = x ^ { 2 } ( x - 36 x )$$

Solve for x (complex solution)

$$x=\frac{-6\sqrt{26}i-18}{35}\approx -0.514285714-0.874117631i$$

$$x=1$$

$$x=\frac{-18+6\sqrt{26}i}{35}\approx -0.514285714+0.874117631i$$

Solution Steps

Combine x and -36x to get -35x.

$$x^{2}-36=x^{2}\left(-35\right)x$$

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

$$x^{2}-36=x^{3}\left(-35\right)$$

Subtract x^{3}\left(-35\right) from both sides.

$$x^{2}-36-x^{3}\left(-35\right)=0$$

Multiply -1 and -35 to get 35.

$$x^{2}-36+35x^{3}=0$$

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

$$35x^{3}+x^{2}-36=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -36 and q divides the leading coefficient 35. List all candidates \frac{p}{q}.

$$±\frac{36}{35},±\frac{36}{7},±\frac{36}{5},±36,±\frac{18}{35},±\frac{18}{7},±\frac{18}{5},±18,±\frac{12}{35},±\frac{12}{7},±\frac{12}{5},±12,±\frac{9}{35},±\frac{9}{7},±\frac{9}{5},±9,±\frac{6}{35},±\frac{6}{7},±\frac{6}{5},±6,±\frac{4}{35},±\frac{4}{7},±\frac{4}{5},±4,±\frac{3}{35},±\frac{3}{7},±\frac{3}{5},±3,±\frac{2}{35},±\frac{2}{7},±\frac{2}{5},±2,±\frac{1}{35},±\frac{1}{7},±\frac{1}{5},±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 35x^{3}+x^{2}-36 by x-1 to get 35x^{2}+36x+36. Solve the equation where the result equals to 0.

$$35x^{2}+36x+36=0$$

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 35 for a, 36 for b, and 36 for c in the quadratic formula.

$$x=\frac{-36±\sqrt{36^{2}-4\times 35\times 36}}{2\times 35}$$

Do the calculations.

$$x=\frac{-36±\sqrt{-3744}}{70}$$

Solve the equation 35x^{2}+36x+36=0 when ± is plus and when ± is minus.

$$x=\frac{-6i\sqrt{26}-18}{35} x=\frac{-18+6i\sqrt{26}}{35}$$

List all found solutions.

$$x=1 x=\frac{-6i\sqrt{26}-18}{35} x=\frac{-18+6i\sqrt{26}}{35}$$

Solve for x

$$x=1$$

Solution Steps

Combine x and -36x to get -35x.

$$x^{2}-36=x^{2}\left(-35\right)x$$

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

$$x^{2}-36=x^{3}\left(-35\right)$$

Subtract x^{3}\left(-35\right) from both sides.

$$x^{2}-36-x^{3}\left(-35\right)=0$$

Multiply -1 and -35 to get 35.

$$x^{2}-36+35x^{3}=0$$

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

$$35x^{3}+x^{2}-36=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -36 and q divides the leading coefficient 35. List all candidates \frac{p}{q}.

$$±\frac{36}{35},±\frac{36}{7},±\frac{36}{5},±36,±\frac{18}{35},±\frac{18}{7},±\frac{18}{5},±18,±\frac{12}{35},±\frac{12}{7},±\frac{12}{5},±12,±\frac{9}{35},±\frac{9}{7},±\frac{9}{5},±9,±\frac{6}{35},±\frac{6}{7},±\frac{6}{5},±6,±\frac{4}{35},±\frac{4}{7},±\frac{4}{5},±4,±\frac{3}{35},±\frac{3}{7},±\frac{3}{5},±3,±\frac{2}{35},±\frac{2}{7},±\frac{2}{5},±2,±\frac{1}{35},±\frac{1}{7},±\frac{1}{5},±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 35x^{3}+x^{2}-36 by x-1 to get 35x^{2}+36x+36. Solve the equation where the result equals to 0.

$$35x^{2}+36x+36=0$$

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 35 for a, 36 for b, and 36 for c in the quadratic formula.

$$x=\frac{-36±\sqrt{36^{2}-4\times 35\times 36}}{2\times 35}$$

Do the calculations.

$$x=\frac{-36±\sqrt{-3744}}{70}$$

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.

$$x=1$$