Solve -12x^2+1200x-28800

$$- 12 x ^ { 2 } + 1200 x - 28800$$

Factor

$$12\left(40-x\right)\left(x-60\right)$$

Solution Steps

Factor out 12.

$$12\left(-x^{2}+100x-2400\right)$$

Consider -x^{2}+100x-2400. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-2400. To find a and b, set up a system to be solved.

$$a+b=100 ab=-\left(-2400\right)=2400$$

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 2400.

$$1,2400 2,1200 3,800 4,600 5,480 6,400 8,300 10,240 12,200 15,160 16,150 20,120 24,100 25,96 30,80 32,75 40,60 48,50$$

Calculate the sum for each pair.

$$1+2400=2401 2+1200=1202 3+800=803 4+600=604 5+480=485 6+400=406 8+300=308 10+240=250 12+200=212 15+160=175 16+150=166 20+120=140 24+100=124 25+96=121 30+80=110 32+75=107 40+60=100 48+50=98$$

The solution is the pair that gives sum 100.

$$a=60 b=40$$

Rewrite -x^{2}+100x-2400 as \left(-x^{2}+60x\right)+\left(40x-2400\right).

$$\left(-x^{2}+60x\right)+\left(40x-2400\right)$$

Factor out -x in the first and 40 in the second group.

$$-x\left(x-60\right)+40\left(x-60\right)$$

Factor out common term x-60 by using distributive property.

$$\left(x-60\right)\left(-x+40\right)$$

Rewrite the complete factored expression.

$$12\left(x-60\right)\left(-x+40\right)$$

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.

$$-12x^{2}+1200x-28800=0$$

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.

$$x=\frac{-1200±\sqrt{1200^{2}-4\left(-12\right)\left(-28800\right)}}{2\left(-12\right)}$$

Square 1200.

$$x=\frac{-1200±\sqrt{1440000-4\left(-12\right)\left(-28800\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-1200±\sqrt{1440000+48\left(-28800\right)}}{2\left(-12\right)}$$

Multiply 48 times -28800.

$$x=\frac{-1200±\sqrt{1440000-1382400}}{2\left(-12\right)}$$

Add 1440000 to -1382400.

$$x=\frac{-1200±\sqrt{57600}}{2\left(-12\right)}$$

Take the square root of 57600.

$$x=\frac{-1200±240}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{-1200±240}{-24}$$

Now solve the equation x=\frac{-1200±240}{-24} when ± is plus. Add -1200 to 240.

$$x=-\frac{960}{-24}$$

Divide -960 by -24.

$$x=40$$

Now solve the equation x=\frac{-1200±240}{-24} when ± is minus. Subtract 240 from -1200.

$$x=-\frac{1440}{-24}$$

Divide -1440 by -24.

$$x=60$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 40 for x_{1} and 60 for x_{2}.

$$-12x^{2}+1200x-28800=-12\left(x-40\right)\left(x-60\right)$$

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.

$$x ^ 2 -100x +2400 = 0$$

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

$$r + s = 100 rs = 2400$$

Two numbers r and s sum up to 100 exactly when the average of the two numbers is \frac{1}{2}*100 = 50. 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>

$$r = 50 - u s = 50 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 2400

$$(50 - u) (50 + u) = 2400$$

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

$$2500 - u^2 = 2400$$

Simplify the expression by subtracting 2500 on both sides

$$-u^2 = 2400-2500 = -100$$

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

$$u^2 = 100 u = \pm\sqrt{100} = \pm 10$$

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

$$r =50 - 10 = 40 s = 50 + 10 = 60$$

Evaluate

$$12\left(40-x\right)\left(x-60\right)$$