Solve h(t)=-16t^2+92t+20

$$h ( t ) = - 16 t ^ { 2 } + 92 t + 20$$

Factor

$$-16\left(t-\frac{23-\sqrt{609}}{8}\right)\left(t-\frac{\sqrt{609}+23}{8}\right)$$

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.

$$-16t^{2}+92t+20=0$$

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.

$$t=\frac{-92±\sqrt{92^{2}-4\left(-16\right)\times 20}}{2\left(-16\right)}$$

Square 92.

$$t=\frac{-92±\sqrt{8464-4\left(-16\right)\times 20}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-92±\sqrt{8464+64\times 20}}{2\left(-16\right)}$$

Multiply 64 times 20.

$$t=\frac{-92±\sqrt{8464+1280}}{2\left(-16\right)}$$

Add 8464 to 1280.

$$t=\frac{-92±\sqrt{9744}}{2\left(-16\right)}$$

Take the square root of 9744.

$$t=\frac{-92±4\sqrt{609}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-92±4\sqrt{609}}{-32}$$

Now solve the equation t=\frac{-92±4\sqrt{609}}{-32} when ± is plus. Add -92 to 4\sqrt{609}.

$$t=\frac{4\sqrt{609}-92}{-32}$$

Divide -92+4\sqrt{609} by -32.

$$t=\frac{23-\sqrt{609}}{8}$$

Now solve the equation t=\frac{-92±4\sqrt{609}}{-32} when ± is minus. Subtract 4\sqrt{609} from -92.

$$t=\frac{-4\sqrt{609}-92}{-32}$$

Divide -92-4\sqrt{609} by -32.

$$t=\frac{\sqrt{609}+23}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{23-\sqrt{609}}{8} for x_{1} and \frac{23+\sqrt{609}}{8} for x_{2}.

$$-16t^{2}+92t+20=-16\left(t-\frac{23-\sqrt{609}}{8}\right)\left(t-\frac{\sqrt{609}+23}{8}\right)$$

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.

$$x ^ 2 -\frac{23}{4}x -\frac{5}{4} = 0$$

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

$$r + s = \frac{23}{4} rs = -\frac{5}{4}$$

Two numbers r and s sum up to \frac{23}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{23}{4} = \frac{23}{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>

$$r = \frac{23}{8} - u s = \frac{23}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{4}

$$(\frac{23}{8} - u) (\frac{23}{8} + u) = -\frac{5}{4}$$

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

$$\frac{529}{64} - u^2 = -\frac{5}{4}$$

Simplify the expression by subtracting \frac{529}{64} on both sides

$$-u^2 = -\frac{5}{4}-\frac{529}{64} = -\frac{609}{64}$$

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

$$u^2 = \frac{609}{64} u = \pm\sqrt{\frac{609}{64}} = \pm \frac{\sqrt{609}}{8}$$

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

$$r =\frac{23}{8} - \frac{\sqrt{609}}{8} = -0.210 s = \frac{23}{8} + \frac{\sqrt{609}}{8} = 5.960$$

Evaluate

$$20+92t-16t^{2}$$