Solve 1000x^2-590x-561=0

$$1000 { x }^{ 2 } -590x-561=0$$

Solve for x

$$x=-\frac{51}{100}=-0.51$$

$$x = \frac{11}{10} = 1\frac{1}{10} = 1.1$$

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 1000x^{2}+ax+bx-561. To find a and b, set up a system to be solved.

$$a+b=-590 ab=1000\left(-561\right)=-561000$$

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 -561000.

$$1,-561000 2,-280500 3,-187000 4,-140250 5,-112200 6,-93500 8,-70125 10,-56100 11,-51000 12,-46750 15,-37400 17,-33000 20,-28050 22,-25500 24,-23375 25,-22440 30,-18700 33,-17000 34,-16500 40,-14025 44,-12750 50,-11220 51,-11000 55,-10200 60,-9350 66,-8500 68,-8250 75,-7480 85,-6600 88,-6375 100,-5610 102,-5500 110,-5100 120,-4675 125,-4488 132,-4250 136,-4125 150,-3740 165,-3400 170,-3300 187,-3000 200,-2805 204,-2750 220,-2550 250,-2244 255,-2200 264,-2125 275,-2040 300,-1870 330,-1700 340,-1650 374,-1500 375,-1496 408,-1375 425,-1320 440,-1275 500,-1122 510,-1100 550,-1020 561,-1000 600,-935 660,-850 680,-825 748,-750$$

Calculate the sum for each pair.

$$1-561000=-560999 2-280500=-280498 3-187000=-186997 4-140250=-140246 5-112200=-112195 6-93500=-93494 8-70125=-70117 10-56100=-56090 11-51000=-50989 12-46750=-46738 15-37400=-37385 17-33000=-32983 20-28050=-28030 22-25500=-25478 24-23375=-23351 25-22440=-22415 30-18700=-18670 33-17000=-16967 34-16500=-16466 40-14025=-13985 44-12750=-12706 50-11220=-11170 51-11000=-10949 55-10200=-10145 60-9350=-9290 66-8500=-8434 68-8250=-8182 75-7480=-7405 85-6600=-6515 88-6375=-6287 100-5610=-5510 102-5500=-5398 110-5100=-4990 120-4675=-4555 125-4488=-4363 132-4250=-4118 136-4125=-3989 150-3740=-3590 165-3400=-3235 170-3300=-3130 187-3000=-2813 200-2805=-2605 204-2750=-2546 220-2550=-2330 250-2244=-1994 255-2200=-1945 264-2125=-1861 275-2040=-1765 300-1870=-1570 330-1700=-1370 340-1650=-1310 374-1500=-1126 375-1496=-1121 408-1375=-967 425-1320=-895 440-1275=-835 500-1122=-622 510-1100=-590 550-1020=-470 561-1000=-439 600-935=-335 660-850=-190 680-825=-145 748-750=-2$$

The solution is the pair that gives sum -590.

$$a=-1100 b=510$$

Rewrite 1000x^{2}-590x-561 as \left(1000x^{2}-1100x\right)+\left(510x-561\right).

$$\left(1000x^{2}-1100x\right)+\left(510x-561\right)$$

Factor out 100x in the first and 51 in the second group.

$$100x\left(10x-11\right)+51\left(10x-11\right)$$

Factor out common term 10x-11 by using distributive property.

$$\left(10x-11\right)\left(100x+51\right)$$

To find equation solutions, solve 10x-11=0 and 100x+51=0.

$$x=\frac{11}{10} x=-\frac{51}{100}$$

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.

$$1000x^{2}-590x-561=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1000 for a, -590 for b, and -561 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-590\right)±\sqrt{\left(-590\right)^{2}-4\times 1000\left(-561\right)}}{2\times 1000}$$

Square -590.

$$x=\frac{-\left(-590\right)±\sqrt{348100-4\times 1000\left(-561\right)}}{2\times 1000}$$

Multiply -4 times 1000.

$$x=\frac{-\left(-590\right)±\sqrt{348100-4000\left(-561\right)}}{2\times 1000}$$

Multiply -4000 times -561.

$$x=\frac{-\left(-590\right)±\sqrt{348100+2244000}}{2\times 1000}$$

Add 348100 to 2244000.

$$x=\frac{-\left(-590\right)±\sqrt{2592100}}{2\times 1000}$$

Take the square root of 2592100.

$$x=\frac{-\left(-590\right)±1610}{2\times 1000}$$

The opposite of -590 is 590.

$$x=\frac{590±1610}{2\times 1000}$$

Multiply 2 times 1000.

$$x=\frac{590±1610}{2000}$$

Now solve the equation x=\frac{590±1610}{2000} when ± is plus. Add 590 to 1610.

$$x=\frac{2200}{2000}$$

Reduce the fraction \frac{2200}{2000} to lowest terms by extracting and canceling out 200.

$$x=\frac{11}{10}$$

Now solve the equation x=\frac{590±1610}{2000} when ± is minus. Subtract 1610 from 590.

$$x=-\frac{1020}{2000}$$

Reduce the fraction \frac{-1020}{2000} to lowest terms by extracting and canceling out 20.

$$x=-\frac{51}{100}$$

The equation is now solved.

$$x=\frac{11}{10} x=-\frac{51}{100}$$

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.

$$1000x^{2}-590x-561=0$$

Add 561 to both sides of the equation.

$$1000x^{2}-590x-561-\left(-561\right)=-\left(-561\right)$$

Subtracting -561 from itself leaves 0.

$$1000x^{2}-590x=-\left(-561\right)$$

Subtract -561 from 0.

$$1000x^{2}-590x=561$$

Divide both sides by 1000.

$$\frac{1000x^{2}-590x}{1000}=\frac{561}{1000}$$

Dividing by 1000 undoes the multiplication by 1000.

$$x^{2}+\left(-\frac{590}{1000}\right)x=\frac{561}{1000}$$

Reduce the fraction \frac{-590}{1000} to lowest terms by extracting and canceling out 10.

$$x^{2}-\frac{59}{100}x=\frac{561}{1000}$$

Divide -\frac{59}{100}, the coefficient of the x term, by 2 to get -\frac{59}{200}. Then add the square of -\frac{59}{200} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{59}{100}x+\left(-\frac{59}{200}\right)^{2}=\frac{561}{1000}+\left(-\frac{59}{200}\right)^{2}$$

Square -\frac{59}{200} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{59}{100}x+\frac{3481}{40000}=\frac{561}{1000}+\frac{3481}{40000}$$

Add \frac{561}{1000} to \frac{3481}{40000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{59}{100}x+\frac{3481}{40000}=\frac{25921}{40000}$$

Factor x^{2}-\frac{59}{100}x+\frac{3481}{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}.

$$\left(x-\frac{59}{200}\right)^{2}=\frac{25921}{40000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{59}{200}\right)^{2}}=\sqrt{\frac{25921}{40000}}$$

Simplify.

$$x-\frac{59}{200}=\frac{161}{200} x-\frac{59}{200}=-\frac{161}{200}$$

Add \frac{59}{200} to both sides of the equation.

$$x=\frac{11}{10} x=-\frac{51}{100}$$