Solve h(t)=-16t^2+64t+192.In

$$h ( t ) = - 16 t ^ { 2 } + 64 t + 192 . In$$

Factor

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

Solution Steps

Factor out 16.

$$16\left(-t^{2}+4t+12\right)$$

Consider -t^{2}+4t+12. Factor the expression by grouping. First, the expression needs to be rewritten as -t^{2}+at+bt+12. To find a and b, set up a system to be solved.

$$a+b=4 ab=-12=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Calculate the sum for each pair.

$$-1+12=11 -2+6=4 -3+4=1$$

The solution is the pair that gives sum 4.

$$a=6 b=-2$$

Rewrite -t^{2}+4t+12 as \left(-t^{2}+6t\right)+\left(-2t+12\right).

$$\left(-t^{2}+6t\right)+\left(-2t+12\right)$$

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

$$-t\left(t-6\right)-2\left(t-6\right)$$

Factor out common term t-6 by using distributive property.

$$\left(t-6\right)\left(-t-2\right)$$

Rewrite the complete factored expression.

$$16\left(t-6\right)\left(-t-2\right)$$

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.

$$-16t^{2}+64t+192=0$$

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.

$$t=\frac{-64±\sqrt{64^{2}-4\left(-16\right)\times 192}}{2\left(-16\right)}$$

Square 64.

$$t=\frac{-64±\sqrt{4096-4\left(-16\right)\times 192}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-64±\sqrt{4096+64\times 192}}{2\left(-16\right)}$$

Multiply 64 times 192.

$$t=\frac{-64±\sqrt{4096+12288}}{2\left(-16\right)}$$

Add 4096 to 12288.

$$t=\frac{-64±\sqrt{16384}}{2\left(-16\right)}$$

Take the square root of 16384.

$$t=\frac{-64±128}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-64±128}{-32}$$

Now solve the equation t=\frac{-64±128}{-32} when ± is plus. Add -64 to 128.

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

Divide 64 by -32.

$$t=-2$$

Now solve the equation t=\frac{-64±128}{-32} when ± is minus. Subtract 128 from -64.

$$t=-\frac{192}{-32}$$

Divide -192 by -32.

$$t=6$$

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 6 for x_{2}.

$$-16t^{2}+64t+192=-16\left(t-\left(-2\right)\right)\left(t-6\right)$$

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

$$-16t^{2}+64t+192=-16\left(t+2\right)\left(t-6\right)$$

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.

$$x ^ 2 -4x -12 = 0$$

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

$$r + s = 4 rs = -12$$

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

$$r = 2 - u s = 2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -12

$$(2 - u) (2 + u) = -12$$

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

$$4 - u^2 = -12$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = -12-4 = -16$$

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

$$u^2 = 16 u = \pm\sqrt{16} = \pm 4$$

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

$$r =2 - 4 = -2 s = 2 + 4 = 6$$

Evaluate

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