Solve 15x^2-300x+1800=0

$$15 x ^ { 2 } - 300 x + 1800 = 0$$

Solve for x (complex solution)

$$x=10+2\sqrt{5}i\approx 10+4.472135955i$$

$$x=-2\sqrt{5}i+10\approx 10-4.472135955i$$

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.

$$15x^{2}-300x+1800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, -300 for b, and 1800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-300\right)±\sqrt{\left(-300\right)^{2}-4\times 15\times 1800}}{2\times 15}$$

Square -300.

$$x=\frac{-\left(-300\right)±\sqrt{90000-4\times 15\times 1800}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-300\right)±\sqrt{90000-60\times 1800}}{2\times 15}$$

Multiply -60 times 1800.

$$x=\frac{-\left(-300\right)±\sqrt{90000-108000}}{2\times 15}$$

Add 90000 to -108000.

$$x=\frac{-\left(-300\right)±\sqrt{-18000}}{2\times 15}$$

Take the square root of -18000.

$$x=\frac{-\left(-300\right)±60\sqrt{5}i}{2\times 15}$$

The opposite of -300 is 300.

$$x=\frac{300±60\sqrt{5}i}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{300±60\sqrt{5}i}{30}$$

Now solve the equation x=\frac{300±60\sqrt{5}i}{30} when ± is plus. Add 300 to 60i\sqrt{5}.

$$x=\frac{300+60\sqrt{5}i}{30}$$

Divide 300+60i\sqrt{5} by 30.

$$x=10+2\sqrt{5}i$$

Now solve the equation x=\frac{300±60\sqrt{5}i}{30} when ± is minus. Subtract 60i\sqrt{5} from 300.

$$x=\frac{-60\sqrt{5}i+300}{30}$$

Divide 300-60i\sqrt{5} by 30.

$$x=-2\sqrt{5}i+10$$

The equation is now solved.

$$x=10+2\sqrt{5}i x=-2\sqrt{5}i+10$$

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.

$$15x^{2}-300x+1800=0$$

Subtract 1800 from both sides of the equation.

$$15x^{2}-300x+1800-1800=-1800$$

Subtracting 1800 from itself leaves 0.

$$15x^{2}-300x=-1800$$

Divide both sides by 15.

$$\frac{15x^{2}-300x}{15}=-\frac{1800}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\left(-\frac{300}{15}\right)x=-\frac{1800}{15}$$

Divide -300 by 15.

$$x^{2}-20x=-\frac{1800}{15}$$

Divide -1800 by 15.

$$x^{2}-20x=-120$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-120+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-120+100$$

Add -120 to 100.

$$x^{2}-20x+100=-20$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=-20$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{-20}$$

Simplify.

$$x-10=2\sqrt{5}i x-10=-2\sqrt{5}i$$

Add 10 to both sides of the equation.

$$x=10+2\sqrt{5}i x=-2\sqrt{5}i+10$$

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 15

$$x ^ 2 -20x +120 = 0$$

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

$$r + s = 20 rs = 120$$

Two numbers r and s sum up to 20 exactly when the average of the two numbers is \frac{1}{2}*20 = 10. 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>

$$r = 10 - u s = 10 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 120

$$(10 - u) (10 + u) = 120$$

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

$$100 - u^2 = 120$$

Simplify the expression by subtracting 100 on both sides

$$-u^2 = 120-100 = 20$$

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

$$u^2 = -20 u = \pm\sqrt{-20} = \pm \sqrt{20}i$$

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

$$r =10 - \sqrt{20}i s = 10 + \sqrt{20}i$$