Solve x^2+10000x+100

$$x ^ { 2 } + 10000 x + 100$$

Factor

$$\left(x-\left(-10\sqrt{249999}-5000\right)\right)\left(x-\left(10\sqrt{249999}-5000\right)\right)$$

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.

$$x^{2}+10000x+100=0$$

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.

$$x=\frac{-10000±\sqrt{10000^{2}-4\times 100}}{2}$$

Square 10000.

$$x=\frac{-10000±\sqrt{100000000-4\times 100}}{2}$$

Multiply -4 times 100.

$$x=\frac{-10000±\sqrt{100000000-400}}{2}$$

Add 100000000 to -400.

$$x=\frac{-10000±\sqrt{99999600}}{2}$$

Take the square root of 99999600.

$$x=\frac{-10000±20\sqrt{249999}}{2}$$

Now solve the equation x=\frac{-10000±20\sqrt{249999}}{2} when ± is plus. Add -10000 to 20\sqrt{249999}.

$$x=\frac{20\sqrt{249999}-10000}{2}$$

Divide -10000+20\sqrt{249999} by 2.

$$x=10\sqrt{249999}-5000$$

Now solve the equation x=\frac{-10000±20\sqrt{249999}}{2} when ± is minus. Subtract 20\sqrt{249999} from -10000.

$$x=\frac{-20\sqrt{249999}-10000}{2}$$

Divide -10000-20\sqrt{249999} by 2.

$$x=-10\sqrt{249999}-5000$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5000+10\sqrt{249999} for x_{1} and -5000-10\sqrt{249999} for x_{2}.

$$x^{2}+10000x+100=\left(x-\left(10\sqrt{249999}-5000\right)\right)\left(x-\left(-10\sqrt{249999}-5000\right)\right)$$

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.

$$x ^ 2 +10000x +100 = 0$$

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

$$r + s = -10000 rs = 100$$

Two numbers r and s sum up to -10000 exactly when the average of the two numbers is \frac{1}{2}*-10000 = -5000. 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>

$$r = -5000 - u s = -5000 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 100

$$(-5000 - u) (-5000 + u) = 100$$

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

$$25000000 - u^2 = 100$$

Simplify the expression by subtracting 25000000 on both sides

$$-u^2 = 100-25000000 = -24999900$$

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

$$u^2 = 24999900 u = \pm\sqrt{24999900} = \pm \sqrt{24999900}$$

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

$$r =-5000 - \sqrt{24999900} = -9999.990 s = -5000 + \sqrt{24999900} = -0.010$$

Evaluate

$$x^{2}+10000x+100$$