Solve 2x^2+400x-960000=0

$$2 { x }^{ 2 } +400x-960000=0$$

Solve for x

$$x=-800$$

$$x=600$$

Steps Using Factoring By Grouping

Divide both sides by 2.

$$x^{2}+200x-480000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-480000. To find a and b, set up a system to be solved.

$$a+b=200 ab=1\left(-480000\right)=-480000$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -480000.

$$-1,480000 -2,240000 -3,160000 -4,120000 -5,96000 -6,80000 -8,60000 -10,48000 -12,40000 -15,32000 -16,30000 -20,24000 -24,20000 -25,19200 -30,16000 -32,15000 -40,12000 -48,10000 -50,9600 -60,8000 -64,7500 -75,6400 -80,6000 -96,5000 -100,4800 -120,4000 -125,3840 -128,3750 -150,3200 -160,3000 -192,2500 -200,2400 -240,2000 -250,1920 -256,1875 -300,1600 -320,1500 -375,1280 -384,1250 -400,1200 -480,1000 -500,960 -600,800 -625,768 -640,750$$

Calculate the sum for each pair.

$$-1+480000=479999 -2+240000=239998 -3+160000=159997 -4+120000=119996 -5+96000=95995 -6+80000=79994 -8+60000=59992 -10+48000=47990 -12+40000=39988 -15+32000=31985 -16+30000=29984 -20+24000=23980 -24+20000=19976 -25+19200=19175 -30+16000=15970 -32+15000=14968 -40+12000=11960 -48+10000=9952 -50+9600=9550 -60+8000=7940 -64+7500=7436 -75+6400=6325 -80+6000=5920 -96+5000=4904 -100+4800=4700 -120+4000=3880 -125+3840=3715 -128+3750=3622 -150+3200=3050 -160+3000=2840 -192+2500=2308 -200+2400=2200 -240+2000=1760 -250+1920=1670 -256+1875=1619 -300+1600=1300 -320+1500=1180 -375+1280=905 -384+1250=866 -400+1200=800 -480+1000=520 -500+960=460 -600+800=200 -625+768=143 -640+750=110$$

The solution is the pair that gives sum 200.

$$a=-600 b=800$$

Rewrite x^{2}+200x-480000 as \left(x^{2}-600x\right)+\left(800x-480000\right).

$$\left(x^{2}-600x\right)+\left(800x-480000\right)$$

Factor out x in the first and 800 in the second group.

$$x\left(x-600\right)+800\left(x-600\right)$$

Factor out common term x-600 by using distributive property.

$$\left(x-600\right)\left(x+800\right)$$

To find equation solutions, solve x-600=0 and x+800=0.

$$x=600 x=-800$$

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.

$$2x^{2}+400x-960000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 400 for b, and -960000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-400±\sqrt{400^{2}-4\times 2\left(-960000\right)}}{2\times 2}$$

Square 400.

$$x=\frac{-400±\sqrt{160000-4\times 2\left(-960000\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-400±\sqrt{160000-8\left(-960000\right)}}{2\times 2}$$

Multiply -8 times -960000.

$$x=\frac{-400±\sqrt{160000+7680000}}{2\times 2}$$

Add 160000 to 7680000.

$$x=\frac{-400±\sqrt{7840000}}{2\times 2}$$

Take the square root of 7840000.

$$x=\frac{-400±2800}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-400±2800}{4}$$

Now solve the equation x=\frac{-400±2800}{4} when ± is plus. Add -400 to 2800.

$$x=\frac{2400}{4}$$

Divide 2400 by 4.

$$x=600$$

Now solve the equation x=\frac{-400±2800}{4} when ± is minus. Subtract 2800 from -400.

$$x=-\frac{3200}{4}$$

Divide -3200 by 4.

$$x=-800$$

The equation is now solved.

$$x=600 x=-800$$

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.

$$2x^{2}+400x-960000=0$$

Add 960000 to both sides of the equation.

$$2x^{2}+400x-960000-\left(-960000\right)=-\left(-960000\right)$$

Subtracting -960000 from itself leaves 0.

$$2x^{2}+400x=-\left(-960000\right)$$

Subtract -960000 from 0.

$$2x^{2}+400x=960000$$

Divide both sides by 2.

$$\frac{2x^{2}+400x}{2}=\frac{960000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{400}{2}x=\frac{960000}{2}$$

Divide 400 by 2.

$$x^{2}+200x=\frac{960000}{2}$$

Divide 960000 by 2.

$$x^{2}+200x=480000$$

Divide 200, the coefficient of the x term, by 2 to get 100. Then add the square of 100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+200x+100^{2}=480000+100^{2}$$

Square 100.

$$x^{2}+200x+10000=480000+10000$$

Add 480000 to 10000.

$$x^{2}+200x+10000=490000$$

Factor x^{2}+200x+10000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+100\right)^{2}=490000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+100\right)^{2}}=\sqrt{490000}$$

Simplify.

$$x+100=700 x+100=-700$$

Subtract 100 from both sides of the equation.

$$x=600 x=-800$$