Solve 0=x^2-60x-21600

$$0 = x ^ { 2 } - 60 x - 21600$$

Solve for x

$$x=-120$$

$$x=180$$

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-60x-21600=0$$

To solve the equation, factor x^{2}-60x-21600 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-60 ab=-21600$$

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 -21600.

$$1,-21600 2,-10800 3,-7200 4,-5400 5,-4320 6,-3600 8,-2700 9,-2400 10,-2160 12,-1800 15,-1440 16,-1350 18,-1200 20,-1080 24,-900 25,-864 27,-800 30,-720 32,-675 36,-600 40,-540 45,-480 48,-450 50,-432 54,-400 60,-360 72,-300 75,-288 80,-270 90,-240 96,-225 100,-216 108,-200 120,-180 135,-160 144,-150$$

Calculate the sum for each pair.

$$1-21600=-21599 2-10800=-10798 3-7200=-7197 4-5400=-5396 5-4320=-4315 6-3600=-3594 8-2700=-2692 9-2400=-2391 10-2160=-2150 12-1800=-1788 15-1440=-1425 16-1350=-1334 18-1200=-1182 20-1080=-1060 24-900=-876 25-864=-839 27-800=-773 30-720=-690 32-675=-643 36-600=-564 40-540=-500 45-480=-435 48-450=-402 50-432=-382 54-400=-346 60-360=-300 72-300=-228 75-288=-213 80-270=-190 90-240=-150 96-225=-129 100-216=-116 108-200=-92 120-180=-60 135-160=-25 144-150=-6$$

The solution is the pair that gives sum -60.

$$a=-180 b=120$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-180\right)\left(x+120\right)$$

To find equation solutions, solve x-180=0 and x+120=0.

$$x=180 x=-120$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-60x-21600=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-21600. To find a and b, set up a system to be solved.

$$a+b=-60 ab=1\left(-21600\right)=-21600$$

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 -21600.

$$1,-21600 2,-10800 3,-7200 4,-5400 5,-4320 6,-3600 8,-2700 9,-2400 10,-2160 12,-1800 15,-1440 16,-1350 18,-1200 20,-1080 24,-900 25,-864 27,-800 30,-720 32,-675 36,-600 40,-540 45,-480 48,-450 50,-432 54,-400 60,-360 72,-300 75,-288 80,-270 90,-240 96,-225 100,-216 108,-200 120,-180 135,-160 144,-150$$

Calculate the sum for each pair.

$$1-21600=-21599 2-10800=-10798 3-7200=-7197 4-5400=-5396 5-4320=-4315 6-3600=-3594 8-2700=-2692 9-2400=-2391 10-2160=-2150 12-1800=-1788 15-1440=-1425 16-1350=-1334 18-1200=-1182 20-1080=-1060 24-900=-876 25-864=-839 27-800=-773 30-720=-690 32-675=-643 36-600=-564 40-540=-500 45-480=-435 48-450=-402 50-432=-382 54-400=-346 60-360=-300 72-300=-228 75-288=-213 80-270=-190 90-240=-150 96-225=-129 100-216=-116 108-200=-92 120-180=-60 135-160=-25 144-150=-6$$

The solution is the pair that gives sum -60.

$$a=-180 b=120$$

Rewrite x^{2}-60x-21600 as \left(x^{2}-180x\right)+\left(120x-21600\right).

$$\left(x^{2}-180x\right)+\left(120x-21600\right)$$

Factor out x in the first and 120 in the second group.

$$x\left(x-180\right)+120\left(x-180\right)$$

Factor out common term x-180 by using distributive property.

$$\left(x-180\right)\left(x+120\right)$$

To find equation solutions, solve x-180=0 and x+120=0.

$$x=180 x=-120$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-60x-21600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -60 for b, and -21600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\left(-21600\right)}}{2}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\left(-21600\right)}}{2}$$

Multiply -4 times -21600.

$$x=\frac{-\left(-60\right)±\sqrt{3600+86400}}{2}$$

Add 3600 to 86400.

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

Take the square root of 90000.

$$x=\frac{-\left(-60\right)±300}{2}$$

The opposite of -60 is 60.

$$x=\frac{60±300}{2}$$

Now solve the equation x=\frac{60±300}{2} when ± is plus. Add 60 to 300.

$$x=\frac{360}{2}$$

Divide 360 by 2.

$$x=180$$

Now solve the equation x=\frac{60±300}{2} when ± is minus. Subtract 300 from 60.

$$x=-\frac{240}{2}$$

Divide -240 by 2.

$$x=-120$$

The equation is now solved.

$$x=180 x=-120$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-60x-21600=0$$

Add 21600 to both sides. Anything plus zero gives itself.

$$x^{2}-60x=21600$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=21600+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=21600+900$$

Add 21600 to 900.

$$x^{2}-60x+900=22500$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=22500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{22500}$$

Simplify.

$$x-30=150 x-30=-150$$

Add 30 to both sides of the equation.

$$x=180 x=-120$$