Solve =-4x^2+32x+192

$$= - 4 x ^ { 2 } + 32 x + 192$$

Factor

$$-4\left(x-12\right)\left(x+4\right)$$

Solution Steps

Factor out 4.

$$4\left(-x^{2}+8x+48\right)$$

Consider -x^{2}+8x+48. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+48. To find a and b, set up a system to be solved.

$$a+b=8 ab=-48=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -48.

$$-1,48 -2,24 -3,16 -4,12 -6,8$$

Calculate the sum for each pair.

$$-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2$$

The solution is the pair that gives sum 8.

$$a=12 b=-4$$

Rewrite -x^{2}+8x+48 as \left(-x^{2}+12x\right)+\left(-4x+48\right).

$$\left(-x^{2}+12x\right)+\left(-4x+48\right)$$

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

$$-x\left(x-12\right)-4\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(-x-4\right)$$

Rewrite the complete factored expression.

$$4\left(x-12\right)\left(-x-4\right)$$

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.

$$-4x^{2}+32x+192=0$$

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.

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

Square 32.

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

Multiply -4 times -4.

$$x=\frac{-32±\sqrt{1024+16\times 192}}{2\left(-4\right)}$$

Multiply 16 times 192.

$$x=\frac{-32±\sqrt{1024+3072}}{2\left(-4\right)}$$

Add 1024 to 3072.

$$x=\frac{-32±\sqrt{4096}}{2\left(-4\right)}$$

Take the square root of 4096.

$$x=\frac{-32±64}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-32±64}{-8}$$

Now solve the equation x=\frac{-32±64}{-8} when ± is plus. Add -32 to 64.

$$x=\frac{32}{-8}$$

Divide 32 by -8.

$$x=-4$$

Now solve the equation x=\frac{-32±64}{-8} when ± is minus. Subtract 64 from -32.

$$x=-\frac{96}{-8}$$

Divide -96 by -8.

$$x=12$$

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 12 for x_{2}.

$$-4x^{2}+32x+192=-4\left(x-\left(-4\right)\right)\left(x-12\right)$$

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

$$-4x^{2}+32x+192=-4\left(x+4\right)\left(x-12\right)$$

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.

$$x ^ 2 -8x -48 = 0$$

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

$$r + s = 8 rs = -48$$

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 = -48

$$(4 - u) (4 + u) = -48$$

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

$$16 - u^2 = -48$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = -48-16 = -64$$

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.

$$r =4 - 8 = -4 s = 4 + 8 = 12$$

Evaluate

$$-4\left(x-12\right)\left(x+4\right)$$