Solve 7x^2+6x+1

$$7 { x }^{ 2 } +6x+1$$

Factor

$$7\left(x-\frac{-\sqrt{2}-3}{7}\right)\left(x-\frac{\sqrt{2}-3}{7}\right)$$

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.

$$7x^{2}+6x+1=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\times 7}}{2\times 7}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 7}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-6±\sqrt{36-28}}{2\times 7}$$

Add 36 to -28.

$$x=\frac{-6±\sqrt{8}}{2\times 7}$$

Take the square root of 8.

$$x=\frac{-6±2\sqrt{2}}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-6±2\sqrt{2}}{14}$$

Now solve the equation x=\frac{-6±2\sqrt{2}}{14} when ± is plus. Add -6 to 2\sqrt{2}.

$$x=\frac{2\sqrt{2}-6}{14}$$

Divide -6+2\sqrt{2} by 14.

$$x=\frac{\sqrt{2}-3}{7}$$

Now solve the equation x=\frac{-6±2\sqrt{2}}{14} when ± is minus. Subtract 2\sqrt{2} from -6.

$$x=\frac{-2\sqrt{2}-6}{14}$$

Divide -6-2\sqrt{2} by 14.

$$x=\frac{-\sqrt{2}-3}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3+\sqrt{2}}{7} for x_{1} and \frac{-3-\sqrt{2}}{7} for x_{2}.

$$7x^{2}+6x+1=7\left(x-\frac{\sqrt{2}-3}{7}\right)\left(x-\frac{-\sqrt{2}-3}{7}\right)$$

Evaluate

$$7x^{2}+6x+1$$