Solve 5x^2+9x+1

$$5 { x }^{ 2 } +9x+1$$

Factor

$$5\left(x-\frac{-\sqrt{61}-9}{10}\right)\left(x-\frac{\sqrt{61}-9}{10}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$5x^{2}+9x+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-9±\sqrt{9^{2}-4\times 5}}{2\times 5}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\times 5}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-9±\sqrt{81-20}}{2\times 5}$$

Add 81 to -20.

$$x=\frac{-9±\sqrt{61}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-9±\sqrt{61}}{10}$$

Now solve the equation x=\frac{-9±\sqrt{61}}{10} when ± is plus. Add -9 to \sqrt{61}.

$$x=\frac{\sqrt{61}-9}{10}$$

Now solve the equation x=\frac{-9±\sqrt{61}}{10} when ± is minus. Subtract \sqrt{61} from -9.

$$x=\frac{-\sqrt{61}-9}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-9+\sqrt{61}}{10} for x_{1} and \frac{-9-\sqrt{61}}{10} for x_{2}.

$$5x^{2}+9x+1=5\left(x-\frac{\sqrt{61}-9}{10}\right)\left(x-\frac{-\sqrt{61}-9}{10}\right)$$

Evaluate

$$5x^{2}+9x+1$$