Solve -2t^2+26t+6

$$- 2 t ^ { 2 } + 26 t + 6$$

Factor

$$-2\left(t-\frac{13-\sqrt{181}}{2}\right)\left(t-\frac{\sqrt{181}+13}{2}\right)$$

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.

$$-2t^{2}+26t+6=0$$

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.

$$t=\frac{-26±\sqrt{26^{2}-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Square 26.

$$t=\frac{-26±\sqrt{676-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$t=\frac{-26±\sqrt{676+8\times 6}}{2\left(-2\right)}$$

Multiply 8 times 6.

$$t=\frac{-26±\sqrt{676+48}}{2\left(-2\right)}$$

Add 676 to 48.

$$t=\frac{-26±\sqrt{724}}{2\left(-2\right)}$$

Take the square root of 724.

$$t=\frac{-26±2\sqrt{181}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$t=\frac{-26±2\sqrt{181}}{-4}$$

Now solve the equation t=\frac{-26±2\sqrt{181}}{-4} when ± is plus. Add -26 to 2\sqrt{181}.

$$t=\frac{2\sqrt{181}-26}{-4}$$

Divide -26+2\sqrt{181} by -4.

$$t=\frac{13-\sqrt{181}}{2}$$

Now solve the equation t=\frac{-26±2\sqrt{181}}{-4} when ± is minus. Subtract 2\sqrt{181} from -26.

$$t=\frac{-2\sqrt{181}-26}{-4}$$

Divide -26-2\sqrt{181} by -4.

$$t=\frac{\sqrt{181}+13}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13-\sqrt{181}}{2} for x_{1} and \frac{13+\sqrt{181}}{2} for x_{2}.

$$-2t^{2}+26t+6=-2\left(t-\frac{13-\sqrt{181}}{2}\right)\left(t-\frac{\sqrt{181}+13}{2}\right)$$

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.

$$x ^ 2 -13x -3 = 0$$

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

$$r + s = 13 rs = -3$$

Two numbers r and s sum up to 13 exactly when the average of the two numbers is \frac{1}{2}*13 = \frac{13}{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>

$$r = \frac{13}{2} - u s = \frac{13}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -3

$$(\frac{13}{2} - u) (\frac{13}{2} + u) = -3$$

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

$$\frac{169}{4} - u^2 = -3$$

Simplify the expression by subtracting \frac{169}{4} on both sides

$$-u^2 = -3-\frac{169}{4} = -\frac{181}{4}$$

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

$$u^2 = \frac{181}{4} u = \pm\sqrt{\frac{181}{4}} = \pm \frac{\sqrt{181}}{2}$$

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

$$r =\frac{13}{2} - \frac{\sqrt{181}}{2} = -0.227 s = \frac{13}{2} + \frac{\sqrt{181}}{2} = 13.227$$

Evaluate

$$6+26t-2t^{2}$$