Solve -16x^2+5184x+421

$$- 16 x ^ { 2 } + 5184 x + 421$$

Factor

$$-16\left(x-\left(-\frac{5\sqrt{16813}}{4}+162\right)\right)\left(x-\left(\frac{5\sqrt{16813}}{4}+162\right)\right)$$

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.

$$-16x^{2}+5184x+421=0$$

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.

$$x=\frac{-5184±\sqrt{5184^{2}-4\left(-16\right)\times 421}}{2\left(-16\right)}$$

Square 5184.

$$x=\frac{-5184±\sqrt{26873856-4\left(-16\right)\times 421}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-5184±\sqrt{26873856+64\times 421}}{2\left(-16\right)}$$

Multiply 64 times 421.

$$x=\frac{-5184±\sqrt{26873856+26944}}{2\left(-16\right)}$$

Add 26873856 to 26944.

$$x=\frac{-5184±\sqrt{26900800}}{2\left(-16\right)}$$

Take the square root of 26900800.

$$x=\frac{-5184±40\sqrt{16813}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-5184±40\sqrt{16813}}{-32}$$

Now solve the equation x=\frac{-5184±40\sqrt{16813}}{-32} when ± is plus. Add -5184 to 40\sqrt{16813}.

$$x=\frac{40\sqrt{16813}-5184}{-32}$$

Divide -5184+40\sqrt{16813} by -32.

$$x=-\frac{5\sqrt{16813}}{4}+162$$

Now solve the equation x=\frac{-5184±40\sqrt{16813}}{-32} when ± is minus. Subtract 40\sqrt{16813} from -5184.

$$x=\frac{-40\sqrt{16813}-5184}{-32}$$

Divide -5184-40\sqrt{16813} by -32.

$$x=\frac{5\sqrt{16813}}{4}+162$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 162-\frac{5\sqrt{16813}}{4} for x_{1} and 162+\frac{5\sqrt{16813}}{4} for x_{2}.

$$-16x^{2}+5184x+421=-16\left(x-\left(-\frac{5\sqrt{16813}}{4}+162\right)\right)\left(x-\left(\frac{5\sqrt{16813}}{4}+162\right)\right)$$

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.

$$x ^ 2 -324x -\frac{421}{16} = 0$$

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

$$r + s = 324 rs = -\frac{421}{16}$$

Two numbers r and s sum up to 324 exactly when the average of the two numbers is \frac{1}{2}*324 = 162. 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>

$$r = 162 - u s = 162 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{421}{16}

$$(162 - u) (162 + u) = -\frac{421}{16}$$

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

$$26244 - u^2 = -\frac{421}{16}$$

Simplify the expression by subtracting 26244 on both sides

$$-u^2 = -\frac{421}{16}-26244 = -\frac{420325}{16}$$

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

$$u^2 = \frac{420325}{16} u = \pm\sqrt{\frac{420325}{16}} = \pm \frac{\sqrt{420325}}{4}$$

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

$$r =162 - \frac{\sqrt{420325}}{4} = -0.081 s = 162 + \frac{\sqrt{420325}}{4} = 324.081$$

Evaluate

$$421+5184x-16x^{2}$$