Solve 8x^2-16x-192

$$8 x ^ { 2 } - 16 x - 192$$

Factor

$$8\left(x-6\right)\left(x+4\right)$$

Solution Steps

Factor out 8.

$$8\left(x^{2}-2x-24\right)$$

Consider x^{2}-2x-24. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-24\right)=-24$$

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 -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

Rewrite x^{2}-2x-24 as \left(x^{2}-6x\right)+\left(4x-24\right).

$$\left(x^{2}-6x\right)+\left(4x-24\right)$$

Factor out x in the first and 4 in the second group.

$$x\left(x-6\right)+4\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+4\right)$$

Rewrite the complete factored expression.

$$8\left(x-6\right)\left(x+4\right)$$

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.

$$8x^{2}-16x-192=0$$

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.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 8\left(-192\right)}}{2\times 8}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 8\left(-192\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-16\right)±\sqrt{256-32\left(-192\right)}}{2\times 8}$$

Multiply -32 times -192.

$$x=\frac{-\left(-16\right)±\sqrt{256+6144}}{2\times 8}$$

Add 256 to 6144.

$$x=\frac{-\left(-16\right)±\sqrt{6400}}{2\times 8}$$

Take the square root of 6400.

$$x=\frac{-\left(-16\right)±80}{2\times 8}$$

The opposite of -16 is 16.

$$x=\frac{16±80}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{16±80}{16}$$

Now solve the equation x=\frac{16±80}{16} when ± is plus. Add 16 to 80.

$$x=\frac{96}{16}$$

Divide 96 by 16.

$$x=6$$

Now solve the equation x=\frac{16±80}{16} when ± is minus. Subtract 80 from 16.

$$x=-\frac{64}{16}$$

Divide -64 by 16.

$$x=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and -4 for x_{2}.

$$8x^{2}-16x-192=8\left(x-6\right)\left(x-\left(-4\right)\right)$$

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

$$8x^{2}-16x-192=8\left(x-6\right)\left(x+4\right)$$

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.This is achieved by dividing both sides of the equation by 8

$$x ^ 2 -2x -24 = 0$$

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

$$r + s = 2 rs = -24$$

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

$$r = 1 - u s = 1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -24

$$(1 - u) (1 + u) = -24$$

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

$$1 - u^2 = -24$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -24-1 = -25$$

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

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

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

$$r =1 - 5 = -4 s = 1 + 5 = 6$$

Evaluate

$$8\left(x-6\right)\left(x+4\right)$$