Solve 12x^2-1200x-960=0

$$12 x ^ { 2 } - 1200 x - 960 = 0$$

Solve for x

$$x=2\sqrt{645}+50\approx 100.793700397$$

$$x=50-2\sqrt{645}\approx -0.793700397$$

Steps Using the Quadratic Formula

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.

$$12x^{2}-1200x-960=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -1200 for b, and -960 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1200\right)±\sqrt{\left(-1200\right)^{2}-4\times 12\left(-960\right)}}{2\times 12}$$

Square -1200.

$$x=\frac{-\left(-1200\right)±\sqrt{1440000-4\times 12\left(-960\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-1200\right)±\sqrt{1440000-48\left(-960\right)}}{2\times 12}$$

Multiply -48 times -960.

$$x=\frac{-\left(-1200\right)±\sqrt{1440000+46080}}{2\times 12}$$

Add 1440000 to 46080.

$$x=\frac{-\left(-1200\right)±\sqrt{1486080}}{2\times 12}$$

Take the square root of 1486080.

$$x=\frac{-\left(-1200\right)±48\sqrt{645}}{2\times 12}$$

The opposite of -1200 is 1200.

$$x=\frac{1200±48\sqrt{645}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{1200±48\sqrt{645}}{24}$$

Now solve the equation x=\frac{1200±48\sqrt{645}}{24} when ± is plus. Add 1200 to 48\sqrt{645}.

$$x=\frac{48\sqrt{645}+1200}{24}$$

Divide 1200+48\sqrt{645} by 24.

$$x=2\sqrt{645}+50$$

Now solve the equation x=\frac{1200±48\sqrt{645}}{24} when ± is minus. Subtract 48\sqrt{645} from 1200.

$$x=\frac{1200-48\sqrt{645}}{24}$$

Divide 1200-48\sqrt{645} by 24.

$$x=50-2\sqrt{645}$$

The equation is now solved.

$$x=2\sqrt{645}+50 x=50-2\sqrt{645}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$12x^{2}-1200x-960=0$$

Add 960 to both sides of the equation.

$$12x^{2}-1200x-960-\left(-960\right)=-\left(-960\right)$$

Subtracting -960 from itself leaves 0.

$$12x^{2}-1200x=-\left(-960\right)$$

Subtract -960 from 0.

$$12x^{2}-1200x=960$$

Divide both sides by 12.

$$\frac{12x^{2}-1200x}{12}=\frac{960}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{1200}{12}\right)x=\frac{960}{12}$$

Divide -1200 by 12.

$$x^{2}-100x=\frac{960}{12}$$

Divide 960 by 12.

$$x^{2}-100x=80$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=80+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=80+2500$$

Add 80 to 2500.

$$x^{2}-100x+2500=2580$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=2580$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2580}$$

Simplify.

$$x-50=2\sqrt{645} x-50=-2\sqrt{645}$$

Add 50 to both sides of the equation.

$$x=2\sqrt{645}+50 x=50-2\sqrt{645}$$

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 12

$$x ^ 2 -100x -80 = 0$$

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

$$r + s = 100 rs = -80$$

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 = -80

$$(50 - u) (50 + u) = -80$$

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

$$2500 - u^2 = -80$$

Simplify the expression by subtracting 2500 on both sides

$$-u^2 = -80-2500 = -2580$$

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

$$u^2 = 2580 u = \pm\sqrt{2580} = \pm \sqrt{2580}$$

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

$$r =50 - \sqrt{2580} = -0.794 s = 50 + \sqrt{2580} = 100.794$$