Solve x^2-16x-512

$$x ^ { 2 } - 16 x - 512$$

Factor

$$\left(x-32\right)\left(x+16\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-512. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\left(-512\right)=-512$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -512.

$$1,-512 2,-256 4,-128 8,-64 16,-32$$

Calculate the sum for each pair.

$$1-512=-511 2-256=-254 4-128=-124 8-64=-56 16-32=-16$$

The solution is the pair that gives sum -16.

$$a=-32 b=16$$

Rewrite x^{2}-16x-512 as \left(x^{2}-32x\right)+\left(16x-512\right).

$$\left(x^{2}-32x\right)+\left(16x-512\right)$$

Factor out x in the first and 16 in the second group.

$$x\left(x-32\right)+16\left(x-32\right)$$

Factor out common term x-32 by using distributive property.

$$\left(x-32\right)\left(x+16\right)$$

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.

$$x^{2}-16x-512=0$$

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.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-512\right)}}{2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-512\right)}}{2}$$

Multiply -4 times -512.

$$x=\frac{-\left(-16\right)±\sqrt{256+2048}}{2}$$

Add 256 to 2048.

$$x=\frac{-\left(-16\right)±\sqrt{2304}}{2}$$

Take the square root of 2304.

$$x=\frac{-\left(-16\right)±48}{2}$$

The opposite of -16 is 16.

$$x=\frac{16±48}{2}$$

Now solve the equation x=\frac{16±48}{2} when ± is plus. Add 16 to 48.

$$x=\frac{64}{2}$$

Divide 64 by 2.

$$x=32$$

Now solve the equation x=\frac{16±48}{2} when ± is minus. Subtract 48 from 16.

$$x=-\frac{32}{2}$$

Divide -32 by 2.

$$x=-16$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 32 for x_{1} and -16 for x_{2}.

$$x^{2}-16x-512=\left(x-32\right)\left(x-\left(-16\right)\right)$$

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

$$x^{2}-16x-512=\left(x-32\right)\left(x+16\right)$$

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.

$$x ^ 2 -16x -512 = 0$$

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

$$r + s = 16 rs = -512$$

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.azureedge.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 = -512

$$(8 - u) (8 + u) = -512$$

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

$$64 - u^2 = -512$$

Simplify the expression by subtracting 64 on both sides

$$-u^2 = -512-64 = -576$$

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

$$u^2 = 576 u = \pm\sqrt{576} = \pm 24$$

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

$$r =8 - 24 = -16 s = 8 + 24 = 32$$

Evaluate

$$\left(x-32\right)\left(x+16\right)$$