Solve -x^2-16x-64

$$- x ^ { 2 } - 16 x - 64$$

Factor

$$-\left(x+8\right)^{2}$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-64. To find a and b, set up a system to be solved.

$$a+b=-16 ab=-\left(-64\right)=64$$

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 64.

$$-1,-64 -2,-32 -4,-16 -8,-8$$

Calculate the sum for each pair.

$$-1-64=-65 -2-32=-34 -4-16=-20 -8-8=-16$$

The solution is the pair that gives sum -16.

$$a=-8 b=-8$$

Rewrite -x^{2}-16x-64 as \left(-x^{2}-8x\right)+\left(-8x-64\right).

$$\left(-x^{2}-8x\right)+\left(-8x-64\right)$$

Factor out -x in the first and -8 in the second group.

$$-x\left(x+8\right)-8\left(x+8\right)$$

Factor out common term x+8 by using distributive property.

$$\left(x+8\right)\left(-x-8\right)$$

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.

$$-x^{2}-16x-64=0$$

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.

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

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-1\right)\left(-64\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-16\right)±\sqrt{256+4\left(-64\right)}}{2\left(-1\right)}$$

Multiply 4 times -64.

$$x=\frac{-\left(-16\right)±\sqrt{256-256}}{2\left(-1\right)}$$

Add 256 to -256.

$$x=\frac{-\left(-16\right)±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=\frac{-\left(-16\right)±0}{2\left(-1\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±0}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{16±0}{-2}$$

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

$$-x^{2}-16x-64=-\left(x-\left(-8\right)\right)\left(x-\left(-8\right)\right)$$

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

$$-x^{2}-16x-64=-\left(x+8\right)\left(x+8\right)$$

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.

$$x ^ 2 +16x +64 = 0$$

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

$$r + s = -16 rs = 64$$

Two numbers r and s sum up to -16 exactly when the average of the two numbers is \frac{1}{2}*-16 = -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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -8 - u s = -8 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 64

$$(-8 - u) (-8 + u) = 64$$

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

$$64 - u^2 = 64$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = 64-64 = 0$$

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.

$$r = s = -8$$

Evaluate

$$-\left(x+8\right)^{2}$$