Solve -16t^2+96t+256=

$$- 16 t ^ { 2 } + 96 t + 256 =$$

Factor

$$-16\left(t-8\right)\left(t+2\right)$$

Solution Steps

Factor out 16.

$$16\left(-t^{2}+6t+16\right)$$

Consider -t^{2}+6t+16. Factor the expression by grouping. First, the expression needs to be rewritten as -t^{2}+at+bt+16. To find a and b, set up a system to be solved.

$$a+b=6 ab=-16=-16$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -16.

$$-1,16 -2,8 -4,4$$

Calculate the sum for each pair.

$$-1+16=15 -2+8=6 -4+4=0$$

The solution is the pair that gives sum 6.

$$a=8 b=-2$$

Rewrite -t^{2}+6t+16 as \left(-t^{2}+8t\right)+\left(-2t+16\right).

$$\left(-t^{2}+8t\right)+\left(-2t+16\right)$$

Factor out -t in the first and -2 in the second group.

$$-t\left(t-8\right)-2\left(t-8\right)$$

Factor out common term t-8 by using distributive property.

$$\left(t-8\right)\left(-t-2\right)$$

Rewrite the complete factored expression.

$$16\left(t-8\right)\left(-t-2\right)$$

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.

$$-16t^{2}+96t+256=0$$

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.

$$t=\frac{-96±\sqrt{96^{2}-4\left(-16\right)\times 256}}{2\left(-16\right)}$$

Square 96.

$$t=\frac{-96±\sqrt{9216-4\left(-16\right)\times 256}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-96±\sqrt{9216+64\times 256}}{2\left(-16\right)}$$

Multiply 64 times 256.

$$t=\frac{-96±\sqrt{9216+16384}}{2\left(-16\right)}$$

Add 9216 to 16384.

$$t=\frac{-96±\sqrt{25600}}{2\left(-16\right)}$$

Take the square root of 25600.

$$t=\frac{-96±160}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-96±160}{-32}$$

Now solve the equation t=\frac{-96±160}{-32} when ± is plus. Add -96 to 160.

$$t=\frac{64}{-32}$$

Divide 64 by -32.

$$t=-2$$

Now solve the equation t=\frac{-96±160}{-32} when ± is minus. Subtract 160 from -96.

$$t=-\frac{256}{-32}$$

Divide -256 by -32.

$$t=8$$

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

$$-16t^{2}+96t+256=-16\left(t-\left(-2\right)\right)\left(t-8\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-16t^{2}+96t+256=-16\left(t+2\right)\left(t-8\right)$$

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.

$$x ^ 2 -6x -16 = 0$$

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

$$r + s = 6 rs = -16$$

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 = -16

$$(3 - u) (3 + u) = -16$$

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

$$9 - u^2 = -16$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = -16-9 = -25$$

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

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

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

$$r =3 - 5 = -2 s = 3 + 5 = 8$$

Evaluate

$$-16\left(t-8\right)\left(t+2\right)$$