Solve L(x)=-5x^2+12000x-15000

$$L ( x ) = - 5 x ^ { 2 } + 12000 x - 15000$$

Factor

$$-5\left(x-\left(1200-10\sqrt{14370}\right)\right)\left(x-\left(10\sqrt{14370}+1200\right)\right)$$

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.

$$-5x^{2}+12000x-15000=0$$

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.

$$x=\frac{-12000±\sqrt{12000^{2}-4\left(-5\right)\left(-15000\right)}}{2\left(-5\right)}$$

Square 12000.

$$x=\frac{-12000±\sqrt{144000000-4\left(-5\right)\left(-15000\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-12000±\sqrt{144000000+20\left(-15000\right)}}{2\left(-5\right)}$$

Multiply 20 times -15000.

$$x=\frac{-12000±\sqrt{144000000-300000}}{2\left(-5\right)}$$

Add 144000000 to -300000.

$$x=\frac{-12000±\sqrt{143700000}}{2\left(-5\right)}$$

Take the square root of 143700000.

$$x=\frac{-12000±100\sqrt{14370}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-12000±100\sqrt{14370}}{-10}$$

Now solve the equation x=\frac{-12000±100\sqrt{14370}}{-10} when ± is plus. Add -12000 to 100\sqrt{14370}.

$$x=\frac{100\sqrt{14370}-12000}{-10}$$

Divide -12000+100\sqrt{14370} by -10.

$$x=1200-10\sqrt{14370}$$

Now solve the equation x=\frac{-12000±100\sqrt{14370}}{-10} when ± is minus. Subtract 100\sqrt{14370} from -12000.

$$x=\frac{-100\sqrt{14370}-12000}{-10}$$

Divide -12000-100\sqrt{14370} by -10.

$$x=10\sqrt{14370}+1200$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1200-10\sqrt{14370} for x_{1} and 1200+10\sqrt{14370} for x_{2}.

$$-5x^{2}+12000x-15000=-5\left(x-\left(1200-10\sqrt{14370}\right)\right)\left(x-\left(10\sqrt{14370}+1200\right)\right)$$

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.

$$x ^ 2 -2400x +3000 = 0$$

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

$$r + s = 2400 rs = 3000$$

Two numbers r and s sum up to 2400 exactly when the average of the two numbers is \frac{1}{2}*2400 = 1200. 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>

$$r = 1200 - u s = 1200 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 3000

$$(1200 - u) (1200 + u) = 3000$$

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

$$1440000 - u^2 = 3000$$

Simplify the expression by subtracting 1440000 on both sides

$$-u^2 = 3000-1440000 = -1437000$$

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

$$u^2 = 1437000 u = \pm\sqrt{1437000} = \pm \sqrt{1437000}$$

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

$$r =1200 - \sqrt{1437000} = 1.251 s = 1200 + \sqrt{1437000} = 2398.749$$

Evaluate

$$-5x^{2}+12000x-15000$$