Solve -16t^2+32t+96

$$- 16 t ^ { 2 } + 32 t + 96$$

Factor

$$-16\left(t-\left(1-\sqrt{7}\right)\right)\left(t-\left(\sqrt{7}+1\right)\right)$$

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.

$$-16t^{2}+32t+96=0$$

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.

$$t=\frac{-32±\sqrt{32^{2}-4\left(-16\right)\times 96}}{2\left(-16\right)}$$

Square 32.

$$t=\frac{-32±\sqrt{1024-4\left(-16\right)\times 96}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-32±\sqrt{1024+64\times 96}}{2\left(-16\right)}$$

Multiply 64 times 96.

$$t=\frac{-32±\sqrt{1024+6144}}{2\left(-16\right)}$$

Add 1024 to 6144.

$$t=\frac{-32±\sqrt{7168}}{2\left(-16\right)}$$

Take the square root of 7168.

$$t=\frac{-32±32\sqrt{7}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-32±32\sqrt{7}}{-32}$$

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

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

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

$$t=1-\sqrt{7}$$

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

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

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

$$t=\sqrt{7}+1$$

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

$$-16t^{2}+32t+96=-16\left(t-\left(1-\sqrt{7}\right)\right)\left(t-\left(\sqrt{7}+1\right)\right)$$

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.

$$x ^ 2 -2x -6 = 0$$

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

$$r + s = 2 rs = -6$$

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

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -6

$$(1 - u) (1 + u) = -6$$

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

$$1 - u^2 = -6$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -6-1 = -7$$

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

$$u^2 = 7 u = \pm\sqrt{7} = \pm \sqrt{7}$$

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

$$r =1 - \sqrt{7} = -1.646 s = 1 + \sqrt{7} = 3.646$$

Evaluate

$$96+32t-16t^{2}$$