Solve f(x)=1+x^2+7x

$$f ( x ) = 1 + x ^ { 2 } + 7 x$$

Factor

$$\left(x-\frac{-3\sqrt{5}-7}{2}\right)\left(x-\frac{3\sqrt{5}-7}{2}\right)$$

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.

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

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.

$$x=\frac{-7±\sqrt{7^{2}-4}}{2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4}}{2}$$

Add 49 to -4.

$$x=\frac{-7±\sqrt{45}}{2}$$

Take the square root of 45.

$$x=\frac{-7±3\sqrt{5}}{2}$$

Now solve the equation x=\frac{-7±3\sqrt{5}}{2} when ± is plus. Add -7 to 3\sqrt{5}.

$$x=\frac{3\sqrt{5}-7}{2}$$

Now solve the equation x=\frac{-7±3\sqrt{5}}{2} when ± is minus. Subtract 3\sqrt{5} from -7.

$$x=\frac{-3\sqrt{5}-7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7+3\sqrt{5}}{2} for x_{1} and \frac{-7-3\sqrt{5}}{2} for x_{2}.

$$x^{2}+7x+1=\left(x-\frac{3\sqrt{5}-7}{2}\right)\left(x-\frac{-3\sqrt{5}-7}{2}\right)$$

Evaluate

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