Solve x^2+32x+192

$$x ^ { 2 } + 32 x + 192$$

Factor

$$\left(x+8\right)\left(x+24\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+192. To find a and b, set up a system to be solved.

$$a+b=32 ab=1\times 192=192$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 192.

$$1,192 2,96 3,64 4,48 6,32 8,24 12,16$$

Calculate the sum for each pair.

$$1+192=193 2+96=98 3+64=67 4+48=52 6+32=38 8+24=32 12+16=28$$

The solution is the pair that gives sum 32.

$$a=8 b=24$$

Rewrite x^{2}+32x+192 as \left(x^{2}+8x\right)+\left(24x+192\right).

$$\left(x^{2}+8x\right)+\left(24x+192\right)$$

Factor out x in the first and 24 in the second group.

$$x\left(x+8\right)+24\left(x+8\right)$$

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

$$\left(x+8\right)\left(x+24\right)$$

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.

$$x^{2}+32x+192=0$$

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.

$$x=\frac{-32±\sqrt{32^{2}-4\times 192}}{2}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\times 192}}{2}$$

Multiply -4 times 192.

$$x=\frac{-32±\sqrt{1024-768}}{2}$$

Add 1024 to -768.

$$x=\frac{-32±\sqrt{256}}{2}$$

Take the square root of 256.

$$x=\frac{-32±16}{2}$$

Now solve the equation x=\frac{-32±16}{2} when ± is plus. Add -32 to 16.

$$x=-\frac{16}{2}$$

Divide -16 by 2.

$$x=-8$$

Now solve the equation x=\frac{-32±16}{2} when ± is minus. Subtract 16 from -32.

$$x=-\frac{48}{2}$$

Divide -48 by 2.

$$x=-24$$

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 -24 for x_{2}.

$$x^{2}+32x+192=\left(x-\left(-8\right)\right)\left(x-\left(-24\right)\right)$$

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

$$x^{2}+32x+192=\left(x+8\right)\left(x+24\right)$$

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.

$$x ^ 2 +32x +192 = 0$$

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

$$r + s = -32 rs = 192$$

Two numbers r and s sum up to -32 exactly when the average of the two numbers is \frac{1}{2}*-32 = -16. 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>

$$r = -16 - u s = -16 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 192

$$(-16 - u) (-16 + u) = 192$$

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

$$256 - u^2 = 192$$

Simplify the expression by subtracting 256 on both sides

$$-u^2 = 192-256 = -64$$

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

$$u^2 = 64 u = \pm\sqrt{64} = \pm 8$$

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

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

Evaluate

$$\left(x+8\right)\left(x+24\right)$$