Solve 28x^3-188x^2+287x+98=0

$$28 { x }^{ 3 } -188 { x }^{ 2 } +287x+98=0$$

Solve for x

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

$$x=-\frac{2}{7}\approx -0.285714286$$

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 98 and q divides the leading coefficient 28. List all candidates \frac{p}{q}.

$$±\frac{7}{2},±7,±14,±\frac{49}{2},±49,±98,±\frac{7}{4},±\frac{49}{4},±\frac{1}{2},±1,±2,±\frac{1}{4},±\frac{1}{14},±\frac{1}{7},±\frac{2}{7},±\frac{1}{28}$$

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.

$$x=-\frac{2}{7}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 28x^{3}-188x^{2}+287x+98 by 7\left(x+\frac{2}{7}\right)=7x+2 to get 4x^{2}-28x+49. Solve the equation where the result equals to 0.

$$4x^{2}-28x+49=0$$

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 4 for a, -28 for b, and 49 for c in the quadratic formula.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 4\times 49}}{2\times 4}$$

Do the calculations.

$$x=\frac{28±0}{8}$$

Solutions are the same.

$$x=\frac{7}{2}$$

List all found solutions.

$$x=-\frac{2}{7} x=\frac{7}{2}$$