Solve -5x^2+16x+20

$$- 5 x ^ { 2 } + 16 x + 20$$

Factor

$$-5\left(x-\frac{8-2\sqrt{41}}{5}\right)\left(x-\frac{2\sqrt{41}+8}{5}\right)$$

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.

$$-5x^{2}+16x+20=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-5\right)\times 20}}{2\left(-5\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-5\right)\times 20}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-16±\sqrt{256+20\times 20}}{2\left(-5\right)}$$

Multiply 20 times 20.

$$x=\frac{-16±\sqrt{256+400}}{2\left(-5\right)}$$

Add 256 to 400.

$$x=\frac{-16±\sqrt{656}}{2\left(-5\right)}$$

Take the square root of 656.

$$x=\frac{-16±4\sqrt{41}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-16±4\sqrt{41}}{-10}$$

Now solve the equation x=\frac{-16±4\sqrt{41}}{-10} when ± is plus. Add -16 to 4\sqrt{41}.

$$x=\frac{4\sqrt{41}-16}{-10}$$

Divide -16+4\sqrt{41} by -10.

$$x=\frac{8-2\sqrt{41}}{5}$$

Now solve the equation x=\frac{-16±4\sqrt{41}}{-10} when ± is minus. Subtract 4\sqrt{41} from -16.

$$x=\frac{-4\sqrt{41}-16}{-10}$$

Divide -16-4\sqrt{41} by -10.

$$x=\frac{2\sqrt{41}+8}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{8-2\sqrt{41}}{5} for x_{1} and \frac{8+2\sqrt{41}}{5} for x_{2}.

$$-5x^{2}+16x+20=-5\left(x-\frac{8-2\sqrt{41}}{5}\right)\left(x-\frac{2\sqrt{41}+8}{5}\right)$$

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.

$$x ^ 2 -\frac{16}{5}x -4 = 0$$

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

$$r + s = \frac{16}{5} rs = -4$$

Two numbers r and s sum up to \frac{16}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{16}{5} = \frac{8}{5}. 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>

$$r = \frac{8}{5} - u s = \frac{8}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -4

$$(\frac{8}{5} - u) (\frac{8}{5} + u) = -4$$

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

$$\frac{64}{25} - u^2 = -4$$

Simplify the expression by subtracting \frac{64}{25} on both sides

$$-u^2 = -4-\frac{64}{25} = -\frac{164}{25}$$

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

$$u^2 = \frac{164}{25} u = \pm\sqrt{\frac{164}{25}} = \pm \frac{\sqrt{164}}{5}$$

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

$$r =\frac{8}{5} - \frac{\sqrt{164}}{5} = -0.961 s = \frac{8}{5} + \frac{\sqrt{164}}{5} = 4.161$$

Evaluate

$$20+16x-5x^{2}$$