Solve 3x^4-2x^3-9x^2+4=8-12x

$$3 x ^ { 4 } - 2 x ^ { 3 } - 9 x ^ { 2 } + 4 = 8 - 12 x$$

Solve for x

$$x=-2$$

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

$$x=1$$

Solution Steps

Subtract 8 from both sides.

$$3x^{4}-2x^{3}-9x^{2}+4-8=-12x$$

Subtract 8 from 4 to get -4.

$$3x^{4}-2x^{3}-9x^{2}-4=-12x$$

Add 12x to both sides.

$$3x^{4}-2x^{3}-9x^{2}-4+12x=0$$

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

$$3x^{4}-2x^{3}-9x^{2}+12x-4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{4}{3},±4,±\frac{2}{3},±2,±\frac{1}{3},±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{4}-2x^{3}-9x^{2}+12x-4 by x-1 to get 3x^{3}+x^{2}-8x+4. Solve the equation where the result equals to 0.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 3. List all candidates \frac{p}{q}.

$$±\frac{4}{3},±4,±\frac{2}{3},±2,±\frac{1}{3},±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 3x^{3}+x^{2}-8x+4 by x-1 to get 3x^{2}+4x-4. Solve the equation where the result equals to 0.

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

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 3 for a, 4 for b, and -4 for c in the quadratic formula.

$$x=\frac{-4±\sqrt{4^{2}-4\times 3\left(-4\right)}}{2\times 3}$$

Do the calculations.

$$x=\frac{-4±8}{6}$$

Solve the equation 3x^{2}+4x-4=0 when ± is plus and when ± is minus.

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

List all found solutions.

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