Solve 4a^3-9a^2+5=0

$$4 a ^ { 3 } - 9 a ^ { 2 } + 5 = 0$$

Solve for a

$$a = \frac{\sqrt{105} + 5}{8} \approx 1.905868846$$

$$a=\frac{5-\sqrt{105}}{8}\approx -0.655868846$$

$$a=1$$

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 4. List all candidates \frac{p}{q}.

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

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.

$$a=1$$

By Factor theorem, a-k is a factor of the polynomial for each root k. Divide 4a^{3}-9a^{2}+5 by a-1 to get 4a^{2}-5a-5. Solve the equation where the result equals to 0.

$$4a^{2}-5a-5=0$$

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

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

Do the calculations.

$$a=\frac{5±\sqrt{105}}{8}$$

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

$$a=\frac{5-\sqrt{105}}{8} a=\frac{\sqrt{105}+5}{8}$$

List all found solutions.

$$a=1 a=\frac{5-\sqrt{105}}{8} a=\frac{\sqrt{105}+5}{8}$$