Solve for x
x = \frac{\sqrt{876524629} - 18107}{8230} \approx 1.397224621
x=\frac{-\sqrt{876524629}-18107}{8230}\approx -5.797467635
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12345x^{2}+54321x-99999=0
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=\frac{-54321±\sqrt{54321^{2}-4\times 12345\left(-99999\right)}}{2\times 12345}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 12345 for a, 54321 for b, and -99999 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-54321±\sqrt{2950771041-4\times 12345\left(-99999\right)}}{2\times 12345}
Square 54321.
x=\frac{-54321±\sqrt{2950771041-49380\left(-99999\right)}}{2\times 12345}
Multiply -4 times 12345.
x=\frac{-54321±\sqrt{2950771041+4937950620}}{2\times 12345}
Multiply -49380 times -99999.
x=\frac{-54321±\sqrt{7888721661}}{2\times 12345}
Add 2950771041 to 4937950620.
x=\frac{-54321±3\sqrt{876524629}}{2\times 12345}
Take the square root of 7888721661.
x=\frac{-54321±3\sqrt{876524629}}{24690}
Multiply 2 times 12345.
x=\frac{3\sqrt{876524629}-54321}{24690}
Now solve the equation x=\frac{-54321±3\sqrt{876524629}}{24690} when ± is plus. Add -54321 to 3\sqrt{876524629}.
x=\frac{\sqrt{876524629}-18107}{8230}
Divide -54321+3\sqrt{876524629} by 24690.
x=\frac{-3\sqrt{876524629}-54321}{24690}
Now solve the equation x=\frac{-54321±3\sqrt{876524629}}{24690} when ± is minus. Subtract 3\sqrt{876524629} from -54321.
x=\frac{-\sqrt{876524629}-18107}{8230}
Divide -54321-3\sqrt{876524629} by 24690.
x=\frac{\sqrt{876524629}-18107}{8230} x=\frac{-\sqrt{876524629}-18107}{8230}
The equation is now solved.
12345x^{2}+54321x-99999=0
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.
12345x^{2}+54321x-99999-\left(-99999\right)=-\left(-99999\right)
Add 99999 to both sides of the equation.
12345x^{2}+54321x=-\left(-99999\right)
Subtracting -99999 from itself leaves 0.
12345x^{2}+54321x=99999
Subtract -99999 from 0.
\frac{12345x^{2}+54321x}{12345}=\frac{99999}{12345}
Divide both sides by 12345.
x^{2}+\frac{54321}{12345}x=\frac{99999}{12345}
Dividing by 12345 undoes the multiplication by 12345.
x^{2}+\frac{18107}{4115}x=\frac{99999}{12345}
Reduce the fraction \frac{54321}{12345} to lowest terms by extracting and canceling out 3.
x^{2}+\frac{18107}{4115}x=\frac{33333}{4115}
Reduce the fraction \frac{99999}{12345} to lowest terms by extracting and canceling out 3.
x^{2}+\frac{18107}{4115}x+\left(\frac{18107}{8230}\right)^{2}=\frac{33333}{4115}+\left(\frac{18107}{8230}\right)^{2}
Divide \frac{18107}{4115}, the coefficient of the x term, by 2 to get \frac{18107}{8230}. Then add the square of \frac{18107}{8230} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{18107}{4115}x+\frac{327863449}{67732900}=\frac{33333}{4115}+\frac{327863449}{67732900}
Square \frac{18107}{8230} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{18107}{4115}x+\frac{327863449}{67732900}=\frac{876524629}{67732900}
Add \frac{33333}{4115} to \frac{327863449}{67732900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{18107}{8230}\right)^{2}=\frac{876524629}{67732900}
Factor x^{2}+\frac{18107}{4115}x+\frac{327863449}{67732900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+\frac{18107}{8230}\right)^{2}}=\sqrt{\frac{876524629}{67732900}}
Take the square root of both sides of the equation.
x+\frac{18107}{8230}=\frac{\sqrt{876524629}}{8230} x+\frac{18107}{8230}=-\frac{\sqrt{876524629}}{8230}
Simplify.
x=\frac{\sqrt{876524629}-18107}{8230} x=\frac{-\sqrt{876524629}-18107}{8230}
Subtract \frac{18107}{8230} from both sides of the equation.
x ^ 2 +\frac{18107}{4115}x -\frac{33333}{4115} = 0
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 12345
r + s = -\frac{18107}{4115} rs = -\frac{33333}{4115}
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 = -\frac{18107}{8230} - u s = -\frac{18107}{8230} + u
Two numbers r and s sum up to -\frac{18107}{4115} exactly when the average of the two numbers is \frac{1}{2}*-\frac{18107}{4115} = -\frac{18107}{8230}. 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>
(-\frac{18107}{8230} - u) (-\frac{18107}{8230} + u) = -\frac{33333}{4115}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{33333}{4115}
\frac{268839251}{121919220} - u^2 = -\frac{33333}{4115}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{33333}{4115}-\frac{268839251}{121919220} = \frac{701212931}{609596100}
Simplify the expression by subtracting \frac{268839251}{121919220} on both sides
u^2 = -\frac{701212931}{609596100} u = \pm\sqrt{-\frac{701212931}{609596100}} = \pm \frac{\sqrt{701212931}}{24690}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{18107}{8230} - \frac{\sqrt{701212931}}{24690}i = -5.797 s = -\frac{18107}{8230} + \frac{\sqrt{701212931}}{24690}i = 1.397
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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