Solve -n^2+n+20

$$- n ^ { 2 } + n + 20$$

Factor

$$-\left(n-5\right)\left(n+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -n^{2}+an+bn+20. To find a and b, set up a system to be solved.

$$a+b=1 ab=-20=-20$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -20.

$$-1,20 -2,10 -4,5$$

Calculate the sum for each pair.

$$-1+20=19 -2+10=8 -4+5=1$$

The solution is the pair that gives sum 1.

$$a=5 b=-4$$

Rewrite -n^{2}+n+20 as \left(-n^{2}+5n\right)+\left(-4n+20\right).

$$\left(-n^{2}+5n\right)+\left(-4n+20\right)$$

Factor out -n in the first and -4 in the second group.

$$-n\left(n-5\right)-4\left(n-5\right)$$

Factor out common term n-5 by using distributive property.

$$\left(n-5\right)\left(-n-4\right)$$

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.

$$-n^{2}+n+20=0$$

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.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 20}}{2\left(-1\right)}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\left(-1\right)\times 20}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-1±\sqrt{1+4\times 20}}{2\left(-1\right)}$$

Multiply 4 times 20.

$$n=\frac{-1±\sqrt{1+80}}{2\left(-1\right)}$$

Add 1 to 80.

$$n=\frac{-1±\sqrt{81}}{2\left(-1\right)}$$

Take the square root of 81.

$$n=\frac{-1±9}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{-1±9}{-2}$$

Now solve the equation n=\frac{-1±9}{-2} when ± is plus. Add -1 to 9.

$$n=\frac{8}{-2}$$

Divide 8 by -2.

$$n=-4$$

Now solve the equation n=\frac{-1±9}{-2} when ± is minus. Subtract 9 from -1.

$$n=-\frac{10}{-2}$$

Divide -10 by -2.

$$n=5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -4 for x_{1} and 5 for x_{2}.

$$-n^{2}+n+20=-\left(n-\left(-4\right)\right)\left(n-5\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-n^{2}+n+20=-\left(n+4\right)\left(n-5\right)$$

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.

$$x ^ 2 -1x -20 = 0$$

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

$$r + s = 1 rs = -20$$

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

$$r = \frac{1}{2} - u s = \frac{1}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -20

$$(\frac{1}{2} - u) (\frac{1}{2} + u) = -20$$

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

$$\frac{1}{4} - u^2 = -20$$

Simplify the expression by subtracting \frac{1}{4} on both sides

$$-u^2 = -20-\frac{1}{4} = -\frac{81}{4}$$

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

$$u^2 = \frac{81}{4} u = \pm\sqrt{\frac{81}{4}} = \pm \frac{9}{2}$$

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

$$r =\frac{1}{2} - \frac{9}{2} = -4 s = \frac{1}{2} + \frac{9}{2} = 5$$

Evaluate

$$-\left(n-5\right)\left(n+4\right)$$