Solve 25t^2-16t+192

$$25 t ^ { 2 } - 16 t + 192$$

Evaluate

$$25t^{2}-16t+192$$

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 25

$$x ^ 2 -\frac{16}{25}x +\frac{192}{25} = 0$$

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

$$r + s = \frac{16}{25} rs = \frac{192}{25}$$

Two numbers r and s sum up to \frac{16}{25} exactly when the average of the two numbers is \frac{1}{2}*\frac{16}{25} = \frac{8}{25}. 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>

$$r = \frac{8}{25} - u s = \frac{8}{25} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{192}{25}

$$(\frac{8}{25} - u) (\frac{8}{25} + u) = \frac{192}{25}$$

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

$$\frac{64}{625} - u^2 = \frac{192}{25}$$

Simplify the expression by subtracting \frac{64}{625} on both sides

$$-u^2 = \frac{192}{25}-\frac{64}{625} = \frac{4736}{625}$$

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

$$u^2 = -\frac{4736}{625} u = \pm\sqrt{-\frac{4736}{625}} = \pm \frac{\sqrt{4736}}{25}i$$

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

$$r =\frac{8}{25} - \frac{\sqrt{4736}}{25}i = 0.320 - 2.753i s = \frac{8}{25} + \frac{\sqrt{4736}}{25}i = 0.320 + 2.753i$$

Differentiate w.r.t. t

$$50t-16$$

Steps Using Derivative Rule for Sum

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$2\times 25t^{2-1}-16t^{1-1}$$

Multiply 2 times 25.

$$50t^{2-1}-16t^{1-1}$$

Subtract 1 from 2.

$$50t^{1}-16t^{1-1}$$

Subtract 1 from 1.

$$50t^{1}-16t^{0}$$

For any term t, t^{1}=t.

$$50t-16t^{0}$$

For any term t except 0, t^{0}=1.

$$50t-16$$