Solve x^2+77x+113

$${ x }^{ 2 } +77x+113$$

Factor

$$\left(x-\frac{-\sqrt{5477}-77}{2}\right)\left(x-\frac{\sqrt{5477}-77}{2}\right)$$

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.

$$x^{2}+77x+113=0$$

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.

$$x=\frac{-77±\sqrt{77^{2}-4\times 113}}{2}$$

Square 77.

$$x=\frac{-77±\sqrt{5929-4\times 113}}{2}$$

Multiply -4 times 113.

$$x=\frac{-77±\sqrt{5929-452}}{2}$$

Add 5929 to -452.

$$x=\frac{-77±\sqrt{5477}}{2}$$

Now solve the equation x=\frac{-77±\sqrt{5477}}{2} when ± is plus. Add -77 to \sqrt{5477}.

$$x=\frac{\sqrt{5477}-77}{2}$$

Now solve the equation x=\frac{-77±\sqrt{5477}}{2} when ± is minus. Subtract \sqrt{5477} from -77.

$$x=\frac{-\sqrt{5477}-77}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-77+\sqrt{5477}}{2} for x_{1} and \frac{-77-\sqrt{5477}}{2} for x_{2}.

$$x^{2}+77x+113=\left(x-\frac{\sqrt{5477}-77}{2}\right)\left(x-\frac{-\sqrt{5477}-77}{2}\right)$$

Evaluate

$$x^{2}+77x+113$$