Solve 20x^3+44x^2-17x-5=0

$$20 x ^ { 3 } + 44 x ^ { 2 } - 17 x - 5 = 0$$

Solve for x

$$x=-\frac{1}{5}=-0.2$$

$$x=\frac{1}{2}=0.5$$

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

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 -5 and q divides the leading coefficient 20. List all candidates \frac{p}{q}.

$$±\frac{1}{4},±\frac{1}{2},±1,±\frac{5}{4},±\frac{5}{2},±5,±\frac{1}{20},±\frac{1}{10},±\frac{1}{5}$$

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.

$$x=-\frac{1}{5}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 20x^{3}+44x^{2}-17x-5 by 5\left(x+\frac{1}{5}\right)=5x+1 to get 4x^{2}+8x-5. Solve the equation where the result equals to 0.

$$4x^{2}+8x-5=0$$

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, 8 for b, and -5 for c in the quadratic formula.

$$x=\frac{-8±\sqrt{8^{2}-4\times 4\left(-5\right)}}{2\times 4}$$

Do the calculations.

$$x=\frac{-8±12}{8}$$

Solve the equation 4x^{2}+8x-5=0 when ± is plus and when ± is minus.

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

List all found solutions.

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