Solve x^2+762048x+16003008

$$x ^ { 2 } + 762048 x + 16003008$$

Factor

$$\left(x-\left(-1512\sqrt{63497}-381024\right)\right)\left(x-\left(1512\sqrt{63497}-381024\right)\right)$$

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.

$$x^{2}+762048x+16003008=0$$

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.

$$x=\frac{-762048±\sqrt{762048^{2}-4\times 16003008}}{2}$$

Square 762048.

$$x=\frac{-762048±\sqrt{580717154304-4\times 16003008}}{2}$$

Multiply -4 times 16003008.

$$x=\frac{-762048±\sqrt{580717154304-64012032}}{2}$$

Add 580717154304 to -64012032.

$$x=\frac{-762048±\sqrt{580653142272}}{2}$$

Take the square root of 580653142272.

$$x=\frac{-762048±3024\sqrt{63497}}{2}$$

Now solve the equation x=\frac{-762048±3024\sqrt{63497}}{2} when ± is plus. Add -762048 to 3024\sqrt{63497}.

$$x=\frac{3024\sqrt{63497}-762048}{2}$$

Divide -762048+3024\sqrt{63497} by 2.

$$x=1512\sqrt{63497}-381024$$

Now solve the equation x=\frac{-762048±3024\sqrt{63497}}{2} when ± is minus. Subtract 3024\sqrt{63497} from -762048.

$$x=\frac{-3024\sqrt{63497}-762048}{2}$$

Divide -762048-3024\sqrt{63497} by 2.

$$x=-1512\sqrt{63497}-381024$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -381024+1512\sqrt{63497} for x_{1} and -381024-1512\sqrt{63497} for x_{2}.

$$x^{2}+762048x+16003008=\left(x-\left(1512\sqrt{63497}-381024\right)\right)\left(x-\left(-1512\sqrt{63497}-381024\right)\right)$$

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.

$$x ^ 2 +762048x [phone] = 0$$

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

$$r + s = -762048 rs = 16003008$$

Two numbers r and s sum up to -762048 exactly when the average of the two numbers is \frac{1}{2}*-762048 = -381024. 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>

$$r = -381024 - u s = -381024 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 16003008

$$(-381024 - u) (-381024 + u) = 16003008$$

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

$$224142336 - u^2 = 16003008$$

Simplify the expression by subtracting 224142336 on both sides

$$-u^2 = 16003008-224142336 = -208139328$$

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

$$u^2 = 208139328 u = \pm\sqrt{208139328} = \pm \sqrt{208139328}$$

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

$$r =-381024 - \sqrt{208139328} = -762026.999 s = -381024 + \sqrt{208139328} = -21.001$$

Evaluate

$$x^{2}+762048x+16003008$$