Solve 6left(x^2right)^2-5x^2x-5x-6=0

$$6 { \left( { x }^{ 2 } \right) }^{ 2 } -5 { x }^{ 2 } x-5x-6=0$$

Solve for x

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

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Solution Steps

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

$$6x^{4}-5xx^{2}-5x-6=0$$

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 6. List all candidates \frac{p}{q}.

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

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 -5xx^{2}-5x+6x^{4}-6 by 3\left(x+\frac{2}{3}\right)=3x+2 to get 2x^{3}-3x^{2}+2x-3. Solve the equation where the result equals to 0.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -3 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±\frac{3}{2},±3,±\frac{1}{2},±1$$

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.

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

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

$$x^{2}+1=0$$

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 1 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{0±\sqrt{-4}}{2}$$

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.

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

Solve for x (complex solution)

$$x=i$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$x=-i$$

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