Solve 2x^2-60x-3600=0

$$2 x ^ { 2 } - 60 x - 3600 = 0$$

Solve for x

$$x=-30$$

$$x=60$$

Steps Using Factoring By Grouping

Divide both sides by 2.

$$x^{2}-30x-1800=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-1800. To find a and b, set up a system to be solved.

$$a+b=-30 ab=1\left(-1800\right)=-1800$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1800.

$$1,-1800 2,-900 3,-600 4,-450 5,-360 6,-300 8,-225 9,-200 10,-180 12,-150 15,-120 18,-100 20,-90 24,-75 25,-72 30,-60 36,-50 40,-45$$

Calculate the sum for each pair.

$$1-1800=-1799 2-900=-898 3-600=-597 4-450=-446 5-360=-355 6-300=-294 8-225=-217 9-200=-191 10-180=-170 12-150=-138 15-120=-105 18-100=-82 20-90=-70 24-75=-51 25-72=-47 30-60=-30 36-50=-14 40-45=-5$$

The solution is the pair that gives sum -30.

$$a=-60 b=30$$

Rewrite x^{2}-30x-1800 as \left(x^{2}-60x\right)+\left(30x-1800\right).

$$\left(x^{2}-60x\right)+\left(30x-1800\right)$$

Factor out x in the first and 30 in the second group.

$$x\left(x-60\right)+30\left(x-60\right)$$

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

$$\left(x-60\right)\left(x+30\right)$$

To find equation solutions, solve x-60=0 and x+30=0.

$$x=60 x=-30$$

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.

$$2x^{2}-60x-3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -60 for b, and -3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 2\left(-3600\right)}}{2\times 2}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 2\left(-3600\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-60\right)±\sqrt{3600-8\left(-3600\right)}}{2\times 2}$$

Multiply -8 times -3600.

$$x=\frac{-\left(-60\right)±\sqrt{3600+28800}}{2\times 2}$$

Add 3600 to 28800.

$$x=\frac{-\left(-60\right)±\sqrt{32400}}{2\times 2}$$

Take the square root of 32400.

$$x=\frac{-\left(-60\right)±180}{2\times 2}$$

The opposite of -60 is 60.

$$x=\frac{60±180}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{60±180}{4}$$

Now solve the equation x=\frac{60±180}{4} when ± is plus. Add 60 to 180.

$$x=\frac{240}{4}$$

Divide 240 by 4.

$$x=60$$

Now solve the equation x=\frac{60±180}{4} when ± is minus. Subtract 180 from 60.

$$x=-\frac{120}{4}$$

Divide -120 by 4.

$$x=-30$$

The equation is now solved.

$$x=60 x=-30$$

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.

$$2x^{2}-60x-3600=0$$

Add 3600 to both sides of the equation.

$$2x^{2}-60x-3600-\left(-3600\right)=-\left(-3600\right)$$

Subtracting -3600 from itself leaves 0.

$$2x^{2}-60x=-\left(-3600\right)$$

Subtract -3600 from 0.

$$2x^{2}-60x=3600$$

Divide both sides by 2.

$$\frac{2x^{2}-60x}{2}=\frac{3600}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{60}{2}\right)x=\frac{3600}{2}$$

Divide -60 by 2.

$$x^{2}-30x=\frac{3600}{2}$$

Divide 3600 by 2.

$$x^{2}-30x=1800$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=1800+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=1800+225$$

Add 1800 to 225.

$$x^{2}-30x+225=2025$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=2025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{2025}$$

Simplify.

$$x-15=45 x-15=-45$$

Add 15 to both sides of the equation.

$$x=60 x=-30$$

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 2

$$x ^ 2 -30x -1800 = 0$$

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

$$r + s = 30 rs = -1800$$

Two numbers r and s sum up to 30 exactly when the average of the two numbers is \frac{1}{2}*30 = 15. 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>

$$r = 15 - u s = 15 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1800

$$(15 - u) (15 + u) = -1800$$

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

$$225 - u^2 = -1800$$

Simplify the expression by subtracting 225 on both sides

$$-u^2 = -1800-225 = -2025$$

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

$$u^2 = 2025 u = \pm\sqrt{2025} = \pm 45$$

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

$$r =15 - 45 = -30 s = 15 + 45 = 60$$