Solve x^2+91x+110

$$x ^ { 2 } + 91 x + 110$$

Factor

$$\left(x-\frac{-\sqrt{7841}-91}{2}\right)\left(x-\frac{\sqrt{7841}-91}{2}\right)$$

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.

$$x^{2}+91x+110=0$$

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.

$$x=\frac{-91±\sqrt{91^{2}-4\times 110}}{2}$$

Square 91.

$$x=\frac{-91±\sqrt{8281-4\times 110}}{2}$$

Multiply -4 times 110.

$$x=\frac{-91±\sqrt{8281-440}}{2}$$

Add 8281 to -440.

$$x=\frac{-91±\sqrt{7841}}{2}$$

Now solve the equation x=\frac{-91±\sqrt{7841}}{2} when ± is plus. Add -91 to \sqrt{7841}.

$$x=\frac{\sqrt{7841}-91}{2}$$

Now solve the equation x=\frac{-91±\sqrt{7841}}{2} when ± is minus. Subtract \sqrt{7841} from -91.

$$x=\frac{-\sqrt{7841}-91}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-91+\sqrt{7841}}{2} for x_{1} and \frac{-91-\sqrt{7841}}{2} for x_{2}.

$$x^{2}+91x+110=\left(x-\frac{\sqrt{7841}-91}{2}\right)\left(x-\frac{-\sqrt{7841}-91}{2}\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +91x +110 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -91 rs = 110$$

Two numbers r and s sum up to -91 exactly when the average of the two numbers is \frac{1}{2}*-91 = -\frac{91}{2}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{91}{2} - u s = -\frac{91}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 110

$$(-\frac{91}{2} - u) (-\frac{91}{2} + u) = 110$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{8281}{4} - u^2 = 110$$

Simplify the expression by subtracting \frac{8281}{4} on both sides

$$-u^2 = 110-\frac{8281}{4} = -\frac{7841}{4}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{7841}{4} u = \pm\sqrt{\frac{7841}{4}} = \pm \frac{\sqrt{7841}}{2}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{91}{2} - \frac{\sqrt{7841}}{2} = -89.775 s = -\frac{91}{2} + \frac{\sqrt{7841}}{2} = -1.225$$

Evaluate

$$x^{2}+91x+110$$