Solve x^2-100x-560000=0

$$x ^ { 2 } - 100 x - 560000 = 0$$

Solve for x

$$x=-700$$

$$x=800$$

Steps Using Factoring

To solve the equation, factor x^{2}-100x-560000 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.

$$a+b=-100 ab=-560000$$

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 -560000.

$$1,-560000 2,-280000 4,-140000 5,-112000 7,-80000 8,-70000 10,-56000 14,-40000 16,-35000 20,-28000 25,-22400 28,-20000 32,-17500 35,-16000 40,-14000 50,-11200 56,-10000 64,-8750 70,-8000 80,-7000 100,-5600 112,-5000 125,-4480 128,-4375 140,-4000 160,-3500 175,-3200 200,-2800 224,-2500 250,-2240 280,-2000 320,-1750 350,-1600 400,-1400 448,-1250 500,-1120 560,-1000 625,-896 640,-875 700,-800$$

Calculate the sum for each pair.

$$1-560000=-559999 2-280000=-279998 4-140000=-139996 5-112000=-111995 7-80000=-79993 8-70000=-69992 10-56000=-55990 14-40000=-39986 16-35000=-34984 20-28000=-27980 25-22400=-22375 28-20000=-19972 32-17500=-17468 35-16000=-15965 40-14000=-13960 50-11200=-11150 56-10000=-9944 64-8750=-8686 70-8000=-7930 80-7000=-6920 100-5600=-5500 112-5000=-4888 125-4480=-4355 128-4375=-4247 140-4000=-3860 160-3500=-3340 175-3200=-3025 200-2800=-2600 224-2500=-2276 250-2240=-1990 280-2000=-1720 320-1750=-1430 350-1600=-1250 400-1400=-1000 448-1250=-802 500-1120=-620 560-1000=-440 625-896=-271 640-875=-235 700-800=-100$$

The solution is the pair that gives sum -100.

$$a=-800 b=700$$

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

$$\left(x-800\right)\left(x+700\right)$$

To find equation solutions, solve x-800=0 and x+700=0.

$$x=800 x=-700$$

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-560000. To find a and b, set up a system to be solved.

$$a+b=-100 ab=1\left(-560000\right)=-560000$$

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 -560000.

$$1,-560000 2,-280000 4,-140000 5,-112000 7,-80000 8,-70000 10,-56000 14,-40000 16,-35000 20,-28000 25,-22400 28,-20000 32,-17500 35,-16000 40,-14000 50,-11200 56,-10000 64,-8750 70,-8000 80,-7000 100,-5600 112,-5000 125,-4480 128,-4375 140,-4000 160,-3500 175,-3200 200,-2800 224,-2500 250,-2240 280,-2000 320,-1750 350,-1600 400,-1400 448,-1250 500,-1120 560,-1000 625,-896 640,-875 700,-800$$

Calculate the sum for each pair.

$$1-560000=-559999 2-280000=-279998 4-140000=-139996 5-112000=-111995 7-80000=-79993 8-70000=-69992 10-56000=-55990 14-40000=-39986 16-35000=-34984 20-28000=-27980 25-22400=-22375 28-20000=-19972 32-17500=-17468 35-16000=-15965 40-14000=-13960 50-11200=-11150 56-10000=-9944 64-8750=-8686 70-8000=-7930 80-7000=-6920 100-5600=-5500 112-5000=-4888 125-4480=-4355 128-4375=-4247 140-4000=-3860 160-3500=-3340 175-3200=-3025 200-2800=-2600 224-2500=-2276 250-2240=-1990 280-2000=-1720 320-1750=-1430 350-1600=-1250 400-1400=-1000 448-1250=-802 500-1120=-620 560-1000=-440 625-896=-271 640-875=-235 700-800=-100$$

The solution is the pair that gives sum -100.

$$a=-800 b=700$$

Rewrite x^{2}-100x-560000 as \left(x^{2}-800x\right)+\left(700x-560000\right).

$$\left(x^{2}-800x\right)+\left(700x-560000\right)$$

Factor out x in the first and 700 in the second group.

$$x\left(x-800\right)+700\left(x-800\right)$$

Factor out common term x-800 by using distributive property.

$$\left(x-800\right)\left(x+700\right)$$

To find equation solutions, solve x-800=0 and x+700=0.

$$x=800 x=-700$$

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.

$$x^{2}-100x-560000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -100 for b, and -560000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\left(-560000\right)}}{2}$$

Square -100.

$$x=\frac{-\left(-100\right)±\sqrt{10000-4\left(-560000\right)}}{2}$$

Multiply -4 times -560000.

$$x=\frac{-\left(-100\right)±\sqrt{10000+2240000}}{2}$$

Add 10000 to 2240000.

$$x=\frac{-\left(-100\right)±\sqrt{2250000}}{2}$$

Take the square root of 2250000.

$$x=\frac{-\left(-100\right)±1500}{2}$$

The opposite of -100 is 100.

$$x=\frac{100±1500}{2}$$

Now solve the equation x=\frac{100±1500}{2} when ± is plus. Add 100 to 1500.

$$x=\frac{1600}{2}$$

Divide 1600 by 2.

$$x=800$$

Now solve the equation x=\frac{100±1500}{2} when ± is minus. Subtract 1500 from 100.

$$x=-\frac{1400}{2}$$

Divide -1400 by 2.

$$x=-700$$

The equation is now solved.

$$x=800 x=-700$$

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.

$$x^{2}-100x-560000=0$$

Add 560000 to both sides of the equation.

$$x^{2}-100x-560000-\left(-560000\right)=-\left(-560000\right)$$

Subtracting -560000 from itself leaves 0.

$$x^{2}-100x=-\left(-560000\right)$$

Subtract -560000 from 0.

$$x^{2}-100x=560000$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=560000+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=560000+2500$$

Add 560000 to 2500.

$$x^{2}-100x+2500=562500$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=562500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{562500}$$

Simplify.

$$x-50=750 x-50=-750$$

Add 50 to both sides of the equation.

$$x=800 x=-700$$

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.

$$x ^ 2 -100x -560000 = 0$$

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

$$r + s = 100 rs = -560000$$

Two numbers r and s sum up to 100 exactly when the average of the two numbers is \frac{1}{2}*100 = 50. 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>

$$r = 50 - u s = 50 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -560000

$$(50 - u) (50 + u) = -560000$$

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

$$2500 - u^2 = -560000$$

Simplify the expression by subtracting 2500 on both sides

$$-u^2 = -560000-2500 = -562500$$

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

$$u^2 = 562500 u = \pm\sqrt{562500} = \pm 750$$

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

$$r =50 - 750 = -700 s = 50 + 750 = 800$$