Solve x^2+122x+120

$$x ^ { 2 } + 122 x + 120$$

Factor

$$\left(x-\left(-\sqrt{3601}-61\right)\right)\left(x-\left(\sqrt{3601}-61\right)\right)$$

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.

$$x^{2}+122x+120=0$$

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.

$$x=\frac{-122±\sqrt{122^{2}-4\times 120}}{2}$$

Square 122.

$$x=\frac{-122±\sqrt{14884-4\times 120}}{2}$$

Multiply -4 times 120.

$$x=\frac{-122±\sqrt{14884-480}}{2}$$

Add 14884 to -480.

$$x=\frac{-122±\sqrt{14404}}{2}$$

Take the square root of 14404.

$$x=\frac{-122±2\sqrt{3601}}{2}$$

Now solve the equation x=\frac{-122±2\sqrt{3601}}{2} when ± is plus. Add -122 to 2\sqrt{3601}.

$$x=\frac{2\sqrt{3601}-122}{2}$$

Divide -122+2\sqrt{3601} by 2.

$$x=\sqrt{3601}-61$$

Now solve the equation x=\frac{-122±2\sqrt{3601}}{2} when ± is minus. Subtract 2\sqrt{3601} from -122.

$$x=\frac{-2\sqrt{3601}-122}{2}$$

Divide -122-2\sqrt{3601} by 2.

$$x=-\sqrt{3601}-61$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -61+\sqrt{3601} for x_{1} and -61-\sqrt{3601} for x_{2}.

$$x^{2}+122x+120=\left(x-\left(\sqrt{3601}-61\right)\right)\left(x-\left(-\sqrt{3601}-61\right)\right)$$

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.

$$x ^ 2 +122x +120 = 0$$

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

$$r + s = -122 rs = 120$$

Two numbers r and s sum up to -122 exactly when the average of the two numbers is \frac{1}{2}*-122 = -61. 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>

$$r = -61 - u s = -61 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 120

$$(-61 - u) (-61 + u) = 120$$

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

$$3721 - u^2 = 120$$

Simplify the expression by subtracting 3721 on both sides

$$-u^2 = 120-3721 = -3601$$

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

$$u^2 = 3601 u = \pm\sqrt{3601} = \pm \sqrt{3601}$$

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

$$r =-61 - \sqrt{3601} = -121.008 s = -61 + \sqrt{3601} = -0.992$$

Evaluate

$$x^{2}+122x+120$$