Solve 75N^2-750N+7200=0

$$75 N ^ { 2 } - 750 N + 7200 = 0$$

Solve for N

$$N=5+\sqrt{71}i\approx 5+8.426149773i$$

$$N=-\sqrt{71}i+5\approx 5-8.426149773i$$

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.

$$75N^{2}-750N+7200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 75 for a, -750 for b, and 7200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$N=\frac{-\left(-750\right)±\sqrt{\left(-750\right)^{2}-4\times 75\times 7200}}{2\times 75}$$

Square -750.

$$N=\frac{-\left(-750\right)±\sqrt{562500-4\times 75\times 7200}}{2\times 75}$$

Multiply -4 times 75.

$$N=\frac{-\left(-750\right)±\sqrt{562500-300\times 7200}}{2\times 75}$$

Multiply -300 times 7200.

$$N=\frac{-\left(-750\right)±\sqrt{562500-2160000}}{2\times 75}$$

Add 562500 to -2160000.

$$N=\frac{-\left(-750\right)±\sqrt{-1597500}}{2\times 75}$$

Take the square root of -1597500.

$$N=\frac{-\left(-750\right)±150\sqrt{71}i}{2\times 75}$$

The opposite of -750 is 750.

$$N=\frac{750±150\sqrt{71}i}{2\times 75}$$

Multiply 2 times 75.

$$N=\frac{750±150\sqrt{71}i}{150}$$

Now solve the equation N=\frac{750±150\sqrt{71}i}{150} when ± is plus. Add 750 to 150i\sqrt{71}.

$$N=\frac{750+150\sqrt{71}i}{150}$$

Divide 750+150i\sqrt{71} by 150.

$$N=5+\sqrt{71}i$$

Now solve the equation N=\frac{750±150\sqrt{71}i}{150} when ± is minus. Subtract 150i\sqrt{71} from 750.

$$N=\frac{-150\sqrt{71}i+750}{150}$$

Divide 750-150i\sqrt{71} by 150.

$$N=-\sqrt{71}i+5$$

The equation is now solved.

$$N=5+\sqrt{71}i N=-\sqrt{71}i+5$$

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.

$$75N^{2}-750N+7200=0$$

Subtract 7200 from both sides of the equation.

$$75N^{2}-750N+7200-7200=-7200$$

Subtracting 7200 from itself leaves 0.

$$75N^{2}-750N=-7200$$

Divide both sides by 75.

$$\frac{75N^{2}-750N}{75}=-\frac{7200}{75}$$

Dividing by 75 undoes the multiplication by 75.

$$N^{2}+\left(-\frac{750}{75}\right)N=-\frac{7200}{75}$$

Divide -750 by 75.

$$N^{2}-10N=-\frac{7200}{75}$$

Divide -7200 by 75.

$$N^{2}-10N=-96$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$N^{2}-10N+\left(-5\right)^{2}=-96+\left(-5\right)^{2}$$

Square -5.

$$N^{2}-10N+25=-96+25$$

Add -96 to 25.

$$N^{2}-10N+25=-71$$

Factor N^{2}-10N+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(N-5\right)^{2}=-71$$

Take the square root of both sides of the equation.

$$\sqrt{\left(N-5\right)^{2}}=\sqrt{-71}$$

Simplify.

$$N-5=\sqrt{71}i N-5=-\sqrt{71}i$$

Add 5 to both sides of the equation.

$$N=5+\sqrt{71}i N=-\sqrt{71}i+5$$

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 75

$$x ^ 2 -10x +96 = 0$$

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

$$r + s = 10 rs = 96$$

Two numbers r and s sum up to 10 exactly when the average of the two numbers is \frac{1}{2}*10 = 5. 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>

$$r = 5 - u s = 5 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 96

$$(5 - u) (5 + u) = 96$$

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

$$25 - u^2 = 96$$

Simplify the expression by subtracting 25 on both sides

$$-u^2 = 96-25 = 71$$

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

$$u^2 = -71 u = \pm\sqrt{-71} = \pm \sqrt{71}i$$

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

$$r =5 - \sqrt{71}i s = 5 + \sqrt{71}i$$