Solve 4x^3-70x^2+300x-375=0

$$4 x ^ { 3 } - 70 x ^ { 2 } + 300 x - 375 = 0$$

Solve for x

$$x = \frac{15 - 5 \sqrt{3}}{2} \approx 3.169872981$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$x = \frac{5 \sqrt{3} + 15}{2} \approx 11.830127019$$

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 -375 and q divides the leading coefficient 4. List all candidates \frac{p}{q}.

$$±\frac{375}{4},±\frac{375}{2},±375,±\frac{125}{4},±\frac{125}{2},±125,±\frac{75}{4},±\frac{75}{2},±75,±\frac{25}{4},±\frac{25}{2},±25,±\frac{15}{4},±\frac{15}{2},±15,±\frac{5}{4},±\frac{5}{2},±5,±\frac{3}{4},±\frac{3}{2},±3,±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$x=\frac{5}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 4x^{3}-70x^{2}+300x-375 by 2\left(x-\frac{5}{2}\right)=2x-5 to get 2x^{2}-30x+75. Solve the equation where the result equals to 0.

$$2x^{2}-30x+75=0$$

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, -30 for b, and 75 for c in the quadratic formula.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 2\times 75}}{2\times 2}$$

Do the calculations.

$$x=\frac{30±10\sqrt{3}}{4}$$

Solve the equation 2x^{2}-30x+75=0 when ± is plus and when ± is minus.

$$x=\frac{15-5\sqrt{3}}{2} x=\frac{5\sqrt{3}+15}{2}$$

List all found solutions.

$$x=\frac{5}{2} x=\frac{15-5\sqrt{3}}{2} x=\frac{5\sqrt{3}+15}{2}$$