Solve x^2-400x-4800000=0

$$x ^ { 2 } - 400 x - 4800000 = 0$$

Solve for x

$$x=-2000$$

$$x=2400$$

Steps Using Factoring

To solve the equation, factor x^{2}-400x-4800000 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.

$$a+b=-400 ab=-4800000$$

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 -4800000.

$$1,-4800000 2,-2400000 3,-1600000 4,-1200000 5,-960000 6,-800000 8,-600000 10,-480000 12,-400000 15,-320000 16,-300000 20,-240000 24,-200000 25,-192000 30,-160000 32,-150000 40,-120000 48,-100000 50,-96000 60,-80000 64,-75000 75,-64000 80,-60000 96,-50000 100,-48000 120,-40000 125,-38400 128,-37500 150,-32000 160,-30000 192,-25000 200,-24000 240,-20000 250,-19200 256,-18750 300,-16000 320,-15000 375,-12800 384,-12500 400,-12000 480,-10000 500,-9600 512,-9375 600,-8000 625,-7680 640,-7500 750,-6400 768,-6250 800,-6000 960,-5000 1000,-4800 1200,-4000 1250,-3840 1280,-3750 1500,-3200 1536,-3125 1600,-3000 1875,-2560 1920,-2500 2000,-2400$$

Calculate the sum for each pair.

$$1-4800000=-4799999 2-2400000=-2399998 3-1600000=-1599997 4-1200000=-1199996 5-960000=-959995 6-800000=-799994 8-600000=-599992 10-480000=-479990 12-400000=-399988 15-320000=-319985 16-300000=-299984 20-240000=-239980 24-200000=-199976 25-192000=-191975 30-160000=-159970 32-150000=-149968 40-120000=-119960 48-100000=-99952 50-96000=-95950 60-80000=-79940 64-75000=-74936 75-64000=-63925 80-60000=-59920 96-50000=-49904 100-48000=-47900 120-40000=-39880 125-38400=-38275 128-37500=-37372 150-32000=-31850 160-30000=-29840 192-25000=-24808 200-24000=-23800 240-20000=-19760 250-19200=-18950 256-18750=-18494 300-16000=-15700 320-15000=-14680 375-12800=-12425 384-12500=-12116 400-12000=-11600 480-10000=-9520 500-9600=-9100 512-9375=-8863 600-8000=-7400 625-7680=-7055 640-7500=-6860 750-6400=-5650 768-6250=-5482 800-6000=-5200 960-5000=-4040 1000-4800=-3800 1200-4000=-2800 1250-3840=-2590 1280-3750=-2470 1500-3200=-1700 1536-3125=-1589 1600-3000=-1400 1875-2560=-685 1920-2500=-580 2000-2400=-400$$

The solution is the pair that gives sum -400.

$$a=-2400 b=2000$$

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

$$\left(x-2400\right)\left(x+2000\right)$$

To find equation solutions, solve x-2400=0 and x+2000=0.

$$x=2400 x=-2000$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-4800000. To find a and b, set up a system to be solved.

$$a+b=-400 ab=1\left(-4800000\right)=-4800000$$

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 -4800000.

$$1,-4800000 2,-2400000 3,-1600000 4,-1200000 5,-960000 6,-800000 8,-600000 10,-480000 12,-400000 15,-320000 16,-300000 20,-240000 24,-200000 25,-192000 30,-160000 32,-150000 40,-120000 48,-100000 50,-96000 60,-80000 64,-75000 75,-64000 80,-60000 96,-50000 100,-48000 120,-40000 125,-38400 128,-37500 150,-32000 160,-30000 192,-25000 200,-24000 240,-20000 250,-19200 256,-18750 300,-16000 320,-15000 375,-12800 384,-12500 400,-12000 480,-10000 500,-9600 512,-9375 600,-8000 625,-7680 640,-7500 750,-6400 768,-6250 800,-6000 960,-5000 1000,-4800 1200,-4000 1250,-3840 1280,-3750 1500,-3200 1536,-3125 1600,-3000 1875,-2560 1920,-2500 2000,-2400$$

Calculate the sum for each pair.

$$1-4800000=-4799999 2-2400000=-2399998 3-1600000=-1599997 4-1200000=-1199996 5-960000=-959995 6-800000=-799994 8-600000=-599992 10-480000=-479990 12-400000=-399988 15-320000=-319985 16-300000=-299984 20-240000=-239980 24-200000=-199976 25-192000=-191975 30-160000=-159970 32-150000=-149968 40-120000=-119960 48-100000=-99952 50-96000=-95950 60-80000=-79940 64-75000=-74936 75-64000=-63925 80-60000=-59920 96-50000=-49904 100-48000=-47900 120-40000=-39880 125-38400=-38275 128-37500=-37372 150-32000=-31850 160-30000=-29840 192-25000=-24808 200-24000=-23800 240-20000=-19760 250-19200=-18950 256-18750=-18494 300-16000=-15700 320-15000=-14680 375-12800=-12425 384-12500=-12116 400-12000=-11600 480-10000=-9520 500-9600=-9100 512-9375=-8863 600-8000=-7400 625-7680=-7055 640-7500=-6860 750-6400=-5650 768-6250=-5482 800-6000=-5200 960-5000=-4040 1000-4800=-3800 1200-4000=-2800 1250-3840=-2590 1280-3750=-2470 1500-3200=-1700 1536-3125=-1589 1600-3000=-1400 1875-2560=-685 1920-2500=-580 2000-2400=-400$$

The solution is the pair that gives sum -400.

$$a=-2400 b=2000$$

Rewrite x^{2}-400x-4800000 as \left(x^{2}-2400x\right)+\left(2000x-4800000\right).

$$\left(x^{2}-2400x\right)+\left(2000x-4800000\right)$$

Factor out x in the first and 2000 in the second group.

$$x\left(x-2400\right)+2000\left(x-2400\right)$$

Factor out common term x-2400 by using distributive property.

$$\left(x-2400\right)\left(x+2000\right)$$

To find equation solutions, solve x-2400=0 and x+2000=0.

$$x=2400 x=-2000$$

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.

$$x^{2}-400x-4800000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -400 for b, and -4800000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-400\right)±\sqrt{\left(-400\right)^{2}-4\left(-4800000\right)}}{2}$$

Square -400.

$$x=\frac{-\left(-400\right)±\sqrt{160000-4\left(-4800000\right)}}{2}$$

Multiply -4 times -4800000.

$$x=\frac{-\left(-400\right)±\sqrt{160000+19200000}}{2}$$

Add 160000 to 19200000.

$$x=\frac{-\left(-400\right)±\sqrt{19360000}}{2}$$

Take the square root of 19360000.

$$x=\frac{-\left(-400\right)±4400}{2}$$

The opposite of -400 is 400.

$$x=\frac{400±4400}{2}$$

Now solve the equation x=\frac{400±4400}{2} when ± is plus. Add 400 to 4400.

$$x=\frac{4800}{2}$$

Divide 4800 by 2.

$$x=2400$$

Now solve the equation x=\frac{400±4400}{2} when ± is minus. Subtract 4400 from 400.

$$x=-\frac{4000}{2}$$

Divide -4000 by 2.

$$x=-2000$$

The equation is now solved.

$$x=2400 x=-2000$$

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.

$$x^{2}-400x-4800000=0$$

Add 4800000 to both sides of the equation.

$$x^{2}-400x-4800000-\left(-4800000\right)=-\left(-4800000\right)$$

Subtracting -4800000 from itself leaves 0.

$$x^{2}-400x=-\left(-4800000\right)$$

Subtract -4800000 from 0.

$$x^{2}-400x=4800000$$

Divide -400, the coefficient of the x term, by 2 to get -200. Then add the square of -200 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-400x+\left(-200\right)^{2}=4800000+\left(-200\right)^{2}$$

Square -200.

$$x^{2}-400x+40000=4800000+40000$$

Add 4800000 to 40000.

$$x^{2}-400x+40000=4840000$$

Factor x^{2}-400x+40000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-200\right)^{2}=4840000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-200\right)^{2}}=\sqrt{4840000}$$

Simplify.

$$x-200=2200 x-200=-2200$$

Add 200 to both sides of the equation.

$$x=2400 x=-2000$$

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.

$$x ^ 2 -400x -4800000 = 0$$

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

$$r + s = 400 rs = -4800000$$

Two numbers r and s sum up to 400 exactly when the average of the two numbers is \frac{1}{2}*400 = 200. 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>

$$r = 200 - u s = 200 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -4800000

$$(200 - u) (200 + u) = -4800000$$

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

$$40000 - u^2 = -4800000$$

Simplify the expression by subtracting 40000 on both sides

$$-u^2 = -4800000-40000 = -4840000$$

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

$$u^2 = 4840000 u = \pm\sqrt{4840000} = \pm 2200$$

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

$$r =200 - 2200 = -2000 s = 200 + 2200 = 2400$$