Solve -16t^2+96t+108

$$- 16 t ^ { 2 } + 96 t + 108$$

Factor

$$-16\left(t-\left(-\frac{3\sqrt{7}}{2}+3\right)\right)\left(t-\left(\frac{3\sqrt{7}}{2}+3\right)\right)$$

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.

$$-16t^{2}+96t+108=0$$

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.

$$t=\frac{-96±\sqrt{96^{2}-4\left(-16\right)\times 108}}{2\left(-16\right)}$$

Square 96.

$$t=\frac{-96±\sqrt{9216-4\left(-16\right)\times 108}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-96±\sqrt{9216+64\times 108}}{2\left(-16\right)}$$

Multiply 64 times 108.

$$t=\frac{-96±\sqrt{9216+6912}}{2\left(-16\right)}$$

Add 9216 to 6912.

$$t=\frac{-96±\sqrt{16128}}{2\left(-16\right)}$$

Take the square root of 16128.

$$t=\frac{-96±48\sqrt{7}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-96±48\sqrt{7}}{-32}$$

Now solve the equation t=\frac{-96±48\sqrt{7}}{-32} when ± is plus. Add -96 to 48\sqrt{7}.

$$t=\frac{48\sqrt{7}-96}{-32}$$

Divide -96+48\sqrt{7} by -32.

$$t=-\frac{3\sqrt{7}}{2}+3$$

Now solve the equation t=\frac{-96±48\sqrt{7}}{-32} when ± is minus. Subtract 48\sqrt{7} from -96.

$$t=\frac{-48\sqrt{7}-96}{-32}$$

Divide -96-48\sqrt{7} by -32.

$$t=\frac{3\sqrt{7}}{2}+3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3-\frac{3\sqrt{7}}{2} for x_{1} and 3+\frac{3\sqrt{7}}{2} for x_{2}.

$$-16t^{2}+96t+108=-16\left(t-\left(-\frac{3\sqrt{7}}{2}+3\right)\right)\left(t-\left(\frac{3\sqrt{7}}{2}+3\right)\right)$$

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.

$$x ^ 2 -6x -\frac{27}{4} = 0$$

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

$$r + s = 6 rs = -\frac{27}{4}$$

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-gzdabgg4ehffg0hf.b01.azurefd.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{27}{4}

$$(3 - u) (3 + u) = -\frac{27}{4}$$

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

$$9 - u^2 = -\frac{27}{4}$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = -\frac{27}{4}-9 = -\frac{63}{4}$$

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

$$u^2 = \frac{63}{4} u = \pm\sqrt{\frac{63}{4}} = \pm \frac{\sqrt{63}}{2}$$

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

$$r =3 - \frac{\sqrt{63}}{2} = -0.969 s = 3 + \frac{\sqrt{63}}{2} = 6.969$$

Evaluate

$$108+96t-16t^{2}$$