Solve 16t^2+96t+256

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

Factor

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

Solution Steps

Factor out 16. Polynomial t^{2}+6t+16 is not factored since it does not have any rational roots.

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

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.

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

Square 96.

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

Multiply -4 times 16.

$$t=\frac{-96±\sqrt{9216-64\times 256}}{2\times 16}$$

Multiply -64 times 256.

$$t=\frac{-96±\sqrt{9216-16384}}{2\times 16}$$

Add 9216 to -16384.

$$t=\frac{-96±\sqrt{-7168}}{2\times 16}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.

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

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.This is achieved by dividing both sides of the equation by 16

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

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

$$-u^2 = 16-9 = 7$$

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

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

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

$$r =-3 - \sqrt{7}i s = -3 + \sqrt{7}i$$

Evaluate

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