Solve h(t)=-16t^2+96t+2

$$h ( t ) = - 16 t ^ { 2 } + 96 t + 2$$

Factor

$$-16\left(t-\left(-\frac{\sqrt{146}}{4}+3\right)\right)\left(t-\left(\frac{\sqrt{146}}{4}+3\right)\right)$$

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.

$$-16t^{2}+96t+2=0$$

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.

$$t=\frac{-96±\sqrt{96^{2}-4\left(-16\right)\times 2}}{2\left(-16\right)}$$

Square 96.

$$t=\frac{-96±\sqrt{9216-4\left(-16\right)\times 2}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-96±\sqrt{9216+64\times 2}}{2\left(-16\right)}$$

Multiply 64 times 2.

$$t=\frac{-96±\sqrt{9216+128}}{2\left(-16\right)}$$

Add 9216 to 128.

$$t=\frac{-96±\sqrt{9344}}{2\left(-16\right)}$$

Take the square root of 9344.

$$t=\frac{-96±8\sqrt{146}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-96±8\sqrt{146}}{-32}$$

Now solve the equation t=\frac{-96±8\sqrt{146}}{-32} when ± is plus. Add -96 to 8\sqrt{146}.

$$t=\frac{8\sqrt{146}-96}{-32}$$

Divide -96+8\sqrt{146} by -32.

$$t=-\frac{\sqrt{146}}{4}+3$$

Now solve the equation t=\frac{-96±8\sqrt{146}}{-32} when ± is minus. Subtract 8\sqrt{146} from -96.

$$t=\frac{-8\sqrt{146}-96}{-32}$$

Divide -96-8\sqrt{146} by -32.

$$t=\frac{\sqrt{146}}{4}+3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3-\frac{\sqrt{146}}{4} for x_{1} and 3+\frac{\sqrt{146}}{4} for x_{2}.

$$-16t^{2}+96t+2=-16\left(t-\left(-\frac{\sqrt{146}}{4}+3\right)\right)\left(t-\left(\frac{\sqrt{146}}{4}+3\right)\right)$$

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.

$$x ^ 2 -6x -\frac{1}{8} = 0$$

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

$$r + s = 6 rs = -\frac{1}{8}$$

Two numbers r and s sum up to 6 exactly when the average of the two numbers is \frac{1}{2}*6 = 3. 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>

$$r = 3 - u s = 3 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{1}{8}

$$(3 - u) (3 + u) = -\frac{1}{8}$$

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

$$9 - u^2 = -\frac{1}{8}$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = -\frac{1}{8}-9 = -\frac{73}{8}$$

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

$$u^2 = \frac{73}{8} u = \pm\sqrt{\frac{73}{8}} = \pm \frac{\sqrt{73}}{\sqrt{8}}$$

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

$$r =3 - \frac{\sqrt{73}}{\sqrt{8}} = -0.021 s = 3 + \frac{\sqrt{73}}{\sqrt{8}} = 6.021$$

Evaluate

$$2+96t-16t^{2}$$