Solve 20500x^2-11025x-11025=0

$$20500 x ^ { 2 } - 11025 x - 11025 = 0$$

Solve for x

$$x=-\frac{21}{41}\approx -0.512195122$$

$$x = \frac{21}{20} = 1\frac{1}{20} = 1.05$$

Steps Using Factoring By Grouping

Divide both sides by 25.

$$820x^{2}-441x-441=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 820x^{2}+ax+bx-441. To find a and b, set up a system to be solved.

$$a+b=-441 ab=820\left(-441\right)=-361620$$

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 -361620.

$$1,-361620 2,-180810 3,-120540 4,-90405 5,-72324 6,-60270 7,-51660 9,-40180 10,-36162 12,-30135 14,-25830 15,-24108 18,-20090 20,-18081 21,-17220 28,-12915 30,-12054 35,-10332 36,-10045 41,-8820 42,-8610 45,-8036 49,-7380 60,-6027 63,-5740 70,-5166 82,-4410 84,-4305 90,-4018 98,-3690 105,-3444 123,-2940 126,-2870 140,-2583 147,-2460 164,-2205 180,-2009 196,-1845 205,-1764 210,-1722 245,-1476 246,-1470 252,-1435 287,-1260 294,-1230 315,-1148 369,-980 410,-882 420,-861 441,-820 490,-738 492,-735 574,-630 588,-615$$

Calculate the sum for each pair.

$$1-361620=-361619 2-180810=-180808 3-120540=-120537 4-90405=-90401 5-72324=-72319 6-60270=-60264 7-51660=-51653 9-40180=-40171 10-36162=-36152 12-30135=-30123 14-25830=-25816 15-24108=-24093 18-20090=-20072 20-18081=-18061 21-17220=-17199 28-12915=-12887 30-12054=-12024 35-10332=-10297 36-10045=-10009 41-8820=-8779 42-8610=-8568 45-8036=-7991 49-7380=-7331 60-6027=-5967 63-5740=-5677 70-5166=-5096 82-4410=-4328 84-4305=-4221 90-4018=-3928 98-3690=-3592 105-3444=-3339 123-2940=-2817 126-2870=-2744 140-2583=-2443 147-2460=-2313 164-2205=-2041 180-2009=-1829 196-1845=-1649 205-1764=-1559 210-1722=-1512 245-1476=-1231 246-1470=-1224 252-1435=-1183 287-1260=-973 294-1230=-936 315-1148=-833 369-980=-611 410-882=-472 420-861=-441 441-820=-379 490-738=-248 492-735=-243 574-630=-56 588-615=-27$$

The solution is the pair that gives sum -441.

$$a=-861 b=420$$

Rewrite 820x^{2}-441x-441 as \left(820x^{2}-861x\right)+\left(420x-441\right).

$$\left(820x^{2}-861x\right)+\left(420x-441\right)$$

Factor out 41x in the first and 21 in the second group.

$$41x\left(20x-21\right)+21\left(20x-21\right)$$

Factor out common term 20x-21 by using distributive property.

$$\left(20x-21\right)\left(41x+21\right)$$

To find equation solutions, solve 20x-21=0 and 41x+21=0.

$$x=\frac{21}{20} x=-\frac{21}{41}$$

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.

$$20500x^{2}-11025x-11025=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20500 for a, -11025 for b, and -11025 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11025\right)±\sqrt{\left(-11025\right)^{2}-4\times 20500\left(-11025\right)}}{2\times 20500}$$

Square -11025.

$$x=\frac{-\left(-11025\right)±\sqrt{121550625-4\times 20500\left(-11025\right)}}{2\times 20500}$$

Multiply -4 times 20500.

$$x=\frac{-\left(-11025\right)±\sqrt{121550625-82000\left(-11025\right)}}{2\times 20500}$$

Multiply -82000 times -11025.

$$x=\frac{-\left(-11025\right)±\sqrt{121550625+904050000}}{2\times 20500}$$

Add 121550625 to 904050000.

$$x=\frac{-\left(-11025\right)±\sqrt{1025600625}}{2\times 20500}$$

Take the square root of 1025600625.

$$x=\frac{-\left(-11025\right)±32025}{2\times 20500}$$

The opposite of -11025 is 11025.

$$x=\frac{11025±32025}{2\times 20500}$$

Multiply 2 times 20500.

$$x=\frac{11025±32025}{41000}$$

Now solve the equation x=\frac{11025±32025}{41000} when ± is plus. Add 11025 to 32025.

$$x=\frac{43050}{41000}$$

Reduce the fraction \frac{43050}{41000} to lowest terms by extracting and canceling out 2050.

$$x=\frac{21}{20}$$

Now solve the equation x=\frac{11025±32025}{41000} when ± is minus. Subtract 32025 from 11025.

$$x=-\frac{21000}{41000}$$

Reduce the fraction \frac{-21000}{41000} to lowest terms by extracting and canceling out 1000.

$$x=-\frac{21}{41}$$

The equation is now solved.

$$x=\frac{21}{20} x=-\frac{21}{41}$$

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.

$$20500x^{2}-11025x-11025=0$$

Add 11025 to both sides of the equation.

$$20500x^{2}-11025x-11025-\left(-11025\right)=-\left(-11025\right)$$

Subtracting -11025 from itself leaves 0.

$$20500x^{2}-11025x=-\left(-11025\right)$$

Subtract -11025 from 0.

$$20500x^{2}-11025x=11025$$

Divide both sides by 20500.

$$\frac{20500x^{2}-11025x}{20500}=\frac{11025}{20500}$$

Dividing by 20500 undoes the multiplication by 20500.

$$x^{2}+\left(-\frac{11025}{20500}\right)x=\frac{11025}{20500}$$

Reduce the fraction \frac{-11025}{20500} to lowest terms by extracting and canceling out 25.

$$x^{2}-\frac{441}{820}x=\frac{11025}{20500}$$

Reduce the fraction \frac{11025}{20500} to lowest terms by extracting and canceling out 25.

$$x^{2}-\frac{441}{820}x=\frac{441}{820}$$

Divide -\frac{441}{820}, the coefficient of the x term, by 2 to get -\frac{441}{1640}. Then add the square of -\frac{441}{1640} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{441}{820}x+\left(-\frac{441}{1640}\right)^{2}=\frac{441}{820}+\left(-\frac{441}{1640}\right)^{2}$$

Square -\frac{441}{1640} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{441}{820}x+\frac{194481}{2689600}=\frac{441}{820}+\frac{194481}{2689600}$$

Add \frac{441}{820} to \frac{194481}{2689600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{441}{820}x+\frac{194481}{2689600}=\frac{1640961}{2689600}$$

Factor x^{2}-\frac{441}{820}x+\frac{194481}{2689600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{441}{1640}\right)^{2}=\frac{1640961}{2689600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{441}{1640}\right)^{2}}=\sqrt{\frac{1640961}{2689600}}$$

Simplify.

$$x-\frac{441}{1640}=\frac{1281}{1640} x-\frac{441}{1640}=-\frac{1281}{1640}$$

Add \frac{441}{1640} to both sides of the equation.

$$x=\frac{21}{20} x=-\frac{21}{41}$$

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.This is achieved by dividing both sides of the equation by 20500

$$x ^ 2 -\frac{441}{820}x -\frac{441}{820} = 0$$

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

$$r + s = \frac{441}{820} rs = -\frac{441}{820}$$

Two numbers r and s sum up to \frac{441}{820} exactly when the average of the two numbers is \frac{1}{2}*\frac{441}{820} = \frac{441}{1640}. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{441}{1640} - u s = \frac{441}{1640} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{441}{820}

$$(\frac{441}{1640} - u) (\frac{441}{1640} + u) = -\frac{441}{820}$$

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

$$-\frac{194481}{2689600} - u^2 = -\frac{441}{820}$$

Simplify the expression by subtracting -\frac{194481}{2689600} on both sides

$$-u^2 = -\frac{441}{820}--\frac{194481}{2689600} = \frac{1640961}{2689600}$$

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

$$u^2 = -\frac{1640961}{2689600} u = \pm\sqrt{-\frac{1640961}{2689600}} = \pm \frac{1281}{1640}i$$

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

$$r =\frac{441}{1640} - \frac{1281}{1640}i = -0.512 s = \frac{441}{1640} + \frac{1281}{1640}i = 1.050$$