Solve -2n^2-102n+140

$$- 2 n ^ { 2 } - 102 n + 140$$

Factor

$$-2\left(n-\frac{-\sqrt{2881}-51}{2}\right)\left(n-\frac{\sqrt{2881}-51}{2}\right)$$

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.

$$-2n^{2}-102n+140=0$$

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.

$$n=\frac{-\left(-102\right)±\sqrt{\left(-102\right)^{2}-4\left(-2\right)\times 140}}{2\left(-2\right)}$$

Square -102.

$$n=\frac{-\left(-102\right)±\sqrt{10404-4\left(-2\right)\times 140}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$n=\frac{-\left(-102\right)±\sqrt{10404+8\times 140}}{2\left(-2\right)}$$

Multiply 8 times 140.

$$n=\frac{-\left(-102\right)±\sqrt{10404+1120}}{2\left(-2\right)}$$

Add 10404 to 1120.

$$n=\frac{-\left(-102\right)±\sqrt{11524}}{2\left(-2\right)}$$

Take the square root of 11524.

$$n=\frac{-\left(-102\right)±2\sqrt{2881}}{2\left(-2\right)}$$

The opposite of -102 is 102.

$$n=\frac{102±2\sqrt{2881}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$n=\frac{102±2\sqrt{2881}}{-4}$$

Now solve the equation n=\frac{102±2\sqrt{2881}}{-4} when ± is plus. Add 102 to 2\sqrt{2881}.

$$n=\frac{2\sqrt{2881}+102}{-4}$$

Divide 102+2\sqrt{2881} by -4.

$$n=\frac{-\sqrt{2881}-51}{2}$$

Now solve the equation n=\frac{102±2\sqrt{2881}}{-4} when ± is minus. Subtract 2\sqrt{2881} from 102.

$$n=\frac{102-2\sqrt{2881}}{-4}$$

Divide 102-2\sqrt{2881} by -4.

$$n=\frac{\sqrt{2881}-51}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-51-\sqrt{2881}}{2} for x_{1} and \frac{-51+\sqrt{2881}}{2} for x_{2}.

$$-2n^{2}-102n+140=-2\left(n-\frac{-\sqrt{2881}-51}{2}\right)\left(n-\frac{\sqrt{2881}-51}{2}\right)$$

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.

$$x ^ 2 +51x -70 = 0$$

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

$$r + s = -51 rs = -70$$

Two numbers r and s sum up to -51 exactly when the average of the two numbers is \frac{1}{2}*-51 = -\frac{51}{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>

$$r = -\frac{51}{2} - u s = -\frac{51}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -70

$$(-\frac{51}{2} - u) (-\frac{51}{2} + u) = -70$$

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

$$\frac{2601}{4} - u^2 = -70$$

Simplify the expression by subtracting \frac{2601}{4} on both sides

$$-u^2 = -70-\frac{2601}{4} = -\frac{2881}{4}$$

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

$$u^2 = \frac{2881}{4} u = \pm\sqrt{\frac{2881}{4}} = \pm \frac{\sqrt{2881}}{2}$$

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

$$r =-\frac{51}{2} - \frac{\sqrt{2881}}{2} = -52.337 s = -\frac{51}{2} + \frac{\sqrt{2881}}{2} = 1.337$$

Evaluate

$$140-102n-2n^{2}$$