Solve -n^2-8n-16

$$- n ^ { 2 } - 8 n - 16$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -n^{2}+an+bn-16. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-\left(-16\right)=16$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 16.

$$-1,-16 -2,-8 -4,-4$$

Calculate the sum for each pair.

$$-1-16=-17 -2-8=-10 -4-4=-8$$

The solution is the pair that gives sum -8.

$$a=-4 b=-4$$

Rewrite -n^{2}-8n-16 as \left(-n^{2}-4n\right)+\left(-4n-16\right).

$$\left(-n^{2}-4n\right)+\left(-4n-16\right)$$

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

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

Factor out common term n+4 by using distributive property.

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

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.

$$-n^{2}-8n-16=0$$

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.

$$n=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Square -8.

$$n=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-\left(-8\right)±\sqrt{64+4\left(-16\right)}}{2\left(-1\right)}$$

Multiply 4 times -16.

$$n=\frac{-\left(-8\right)±\sqrt{64-64}}{2\left(-1\right)}$$

Add 64 to -64.

$$n=\frac{-\left(-8\right)±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$n=\frac{-\left(-8\right)±0}{2\left(-1\right)}$$

The opposite of -8 is 8.

$$n=\frac{8±0}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{8±0}{-2}$$

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 -4 for x_{2}.

$$-n^{2}-8n-16=-\left(n-\left(-4\right)\right)\left(n-\left(-4\right)\right)$$

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

$$-n^{2}-8n-16=-\left(n+4\right)\left(n+4\right)$$

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.

$$x ^ 2 +8x +16 = 0$$

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

$$r + s = -8 rs = 16$$

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

$$r = -4 - u s = -4 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 16

$$(-4 - u) (-4 + u) = 16$$

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

$$16 - u^2 = 16$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = 16-16 = 0$$

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

$$u^2 = 0 u = 0$$

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

$$r = s = -4$$

Evaluate

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