Solve -16t^2+60t+6

$$- 16 t ^ { 2 } + 60 t + 6$$

Factor

$$-16\left(t-\frac{15-\sqrt{249}}{8}\right)\left(t-\frac{\sqrt{249}+15}{8}\right)$$

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.

$$-16t^{2}+60t+6=0$$

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.

$$t=\frac{-60±\sqrt{60^{2}-4\left(-16\right)\times 6}}{2\left(-16\right)}$$

Square 60.

$$t=\frac{-60±\sqrt{3600-4\left(-16\right)\times 6}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-60±\sqrt{3600+64\times 6}}{2\left(-16\right)}$$

Multiply 64 times 6.

$$t=\frac{-60±\sqrt{3600+384}}{2\left(-16\right)}$$

Add 3600 to 384.

$$t=\frac{-60±\sqrt{3984}}{2\left(-16\right)}$$

Take the square root of 3984.

$$t=\frac{-60±4\sqrt{249}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-60±4\sqrt{249}}{-32}$$

Now solve the equation t=\frac{-60±4\sqrt{249}}{-32} when ± is plus. Add -60 to 4\sqrt{249}.

$$t=\frac{4\sqrt{249}-60}{-32}$$

Divide -60+4\sqrt{249} by -32.

$$t=\frac{15-\sqrt{249}}{8}$$

Now solve the equation t=\frac{-60±4\sqrt{249}}{-32} when ± is minus. Subtract 4\sqrt{249} from -60.

$$t=\frac{-4\sqrt{249}-60}{-32}$$

Divide -60-4\sqrt{249} by -32.

$$t=\frac{\sqrt{249}+15}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{15-\sqrt{249}}{8} for x_{1} and \frac{15+\sqrt{249}}{8} for x_{2}.

$$-16t^{2}+60t+6=-16\left(t-\frac{15-\sqrt{249}}{8}\right)\left(t-\frac{\sqrt{249}+15}{8}\right)$$

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.

$$x ^ 2 -\frac{15}{4}x -\frac{3}{8} = 0$$

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

$$r + s = \frac{15}{4} rs = -\frac{3}{8}$$

Two numbers r and s sum up to \frac{15}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{15}{4} = \frac{15}{8}. 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>

$$r = \frac{15}{8} - u s = \frac{15}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{3}{8}

$$(\frac{15}{8} - u) (\frac{15}{8} + u) = -\frac{3}{8}$$

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

$$\frac{225}{64} - u^2 = -\frac{3}{8}$$

Simplify the expression by subtracting \frac{225}{64} on both sides

$$-u^2 = -\frac{3}{8}-\frac{225}{64} = -\frac{249}{64}$$

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

$$u^2 = \frac{249}{64} u = \pm\sqrt{\frac{249}{64}} = \pm \frac{\sqrt{249}}{8}$$

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

$$r =\frac{15}{8} - \frac{\sqrt{249}}{8} = -0.097 s = \frac{15}{8} + \frac{\sqrt{249}}{8} = 3.847$$

Evaluate

$$6+60t-16t^{2}$$