Solve 4x^3-18x^2+27=0

$$4 x ^ { 3 } - 18 x ^ { 2 } + 27 = 0$$

Solve for x

$$x = \frac{3 \sqrt{3} + 3}{2} \approx 4.098076211$$

$$x=\frac{3-3\sqrt{3}}{2}\approx -1.098076211$$

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

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 27 and q divides the leading coefficient 4. List all candidates \frac{p}{q}.

$$±\frac{27}{4},±\frac{27}{2},±27,±\frac{9}{4},±\frac{9}{2},±9,±\frac{3}{4},±\frac{3}{2},±3,±\frac{1}{4},±\frac{1}{2},±1$$

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 4x^{3}-18x^{2}+27 by 2\left(x-\frac{3}{2}\right)=2x-3 to get 2x^{2}-6x-9. Solve the equation where the result equals to 0.

$$2x^{2}-6x-9=0$$

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 2 for a, -6 for b, and -9 for c in the quadratic formula.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\left(-9\right)}}{2\times 2}$$

Do the calculations.

$$x=\frac{6±6\sqrt{3}}{4}$$

Solve the equation 2x^{2}-6x-9=0 when ± is plus and when ± is minus.

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

List all found solutions.

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