Solve -16t^2+100t+200

$$- 16 t ^ { 2 } + 100 t + 200$$

Factor

$$-16\left(t-\frac{25-5\sqrt{57}}{8}\right)\left(t-\frac{5\sqrt{57}+25}{8}\right)$$

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.

$$-16t^{2}+100t+200=0$$

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.

$$t=\frac{-100±\sqrt{100^{2}-4\left(-16\right)\times 200}}{2\left(-16\right)}$$

Square 100.

$$t=\frac{-100±\sqrt{10000-4\left(-16\right)\times 200}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-100±\sqrt{10000+64\times 200}}{2\left(-16\right)}$$

Multiply 64 times 200.

$$t=\frac{-100±\sqrt{10000+12800}}{2\left(-16\right)}$$

Add 10000 to 12800.

$$t=\frac{-100±\sqrt{22800}}{2\left(-16\right)}$$

Take the square root of 22800.

$$t=\frac{-100±20\sqrt{57}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-100±20\sqrt{57}}{-32}$$

Now solve the equation t=\frac{-100±20\sqrt{57}}{-32} when ± is plus. Add -100 to 20\sqrt{57}.

$$t=\frac{20\sqrt{57}-100}{-32}$$

Divide -100+20\sqrt{57} by -32.

$$t=\frac{25-5\sqrt{57}}{8}$$

Now solve the equation t=\frac{-100±20\sqrt{57}}{-32} when ± is minus. Subtract 20\sqrt{57} from -100.

$$t=\frac{-20\sqrt{57}-100}{-32}$$

Divide -100-20\sqrt{57} by -32.

$$t=\frac{5\sqrt{57}+25}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{25-5\sqrt{57}}{8} for x_{1} and \frac{25+5\sqrt{57}}{8} for x_{2}.

$$-16t^{2}+100t+200=-16\left(t-\frac{25-5\sqrt{57}}{8}\right)\left(t-\frac{5\sqrt{57}+25}{8}\right)$$

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.

$$x ^ 2 -\frac{25}{4}x -\frac{25}{2} = 0$$

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

$$r + s = \frac{25}{4} rs = -\frac{25}{2}$$

Two numbers r and s sum up to \frac{25}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{25}{4} = \frac{25}{8}. 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>

$$r = \frac{25}{8} - u s = \frac{25}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{25}{2}

$$(\frac{25}{8} - u) (\frac{25}{8} + u) = -\frac{25}{2}$$

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

$$\frac{625}{64} - u^2 = -\frac{25}{2}$$

Simplify the expression by subtracting \frac{625}{64} on both sides

$$-u^2 = -\frac{25}{2}-\frac{625}{64} = -\frac{1425}{64}$$

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

$$u^2 = \frac{1425}{64} u = \pm\sqrt{\frac{1425}{64}} = \pm \frac{\sqrt{1425}}{8}$$

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

$$r =\frac{25}{8} - \frac{\sqrt{1425}}{8} = -1.594 s = \frac{25}{8} + \frac{\sqrt{1425}}{8} = 7.844$$

Evaluate

$$200+100t-16t^{2}$$