Solve for x (complex solution)
x=\frac{19+\sqrt{80423}i}{20}\approx 0.95+14.179474602i
x=\frac{-\sqrt{80423}i+19}{20}\approx 0.95-14.179474602i
Graph
Share
Copied to clipboard
50x^{2}-95x+10098=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{-\left(-95\right)±\sqrt{\left(-95\right)^{2}-4\times 50\times 10098}}{2\times 50}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 50 for a, -95 for b, and 10098 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-95\right)±\sqrt{9025-4\times 50\times 10098}}{2\times 50}
Square -95.
x=\frac{-\left(-95\right)±\sqrt{9025-200\times 10098}}{2\times 50}
Multiply -4 times 50.
x=\frac{-\left(-95\right)±\sqrt{9025-2019600}}{2\times 50}
Multiply -200 times 10098.
x=\frac{-\left(-95\right)±\sqrt{-2010575}}{2\times 50}
Add 9025 to -2019600.
x=\frac{-\left(-95\right)±5\sqrt{80423}i}{2\times 50}
Take the square root of -2010575.
x=\frac{95±5\sqrt{80423}i}{2\times 50}
The opposite of -95 is 95.
x=\frac{95±5\sqrt{80423}i}{100}
Multiply 2 times 50.
x=\frac{95+5\sqrt{80423}i}{100}
Now solve the equation x=\frac{95±5\sqrt{80423}i}{100} when ± is plus. Add 95 to 5i\sqrt{80423}.
x=\frac{19+\sqrt{80423}i}{20}
Divide 95+5i\sqrt{80423} by 100.
x=\frac{-5\sqrt{80423}i+95}{100}
Now solve the equation x=\frac{95±5\sqrt{80423}i}{100} when ± is minus. Subtract 5i\sqrt{80423} from 95.
x=\frac{-\sqrt{80423}i+19}{20}
Divide 95-5i\sqrt{80423} by 100.
x=\frac{19+\sqrt{80423}i}{20} x=\frac{-\sqrt{80423}i+19}{20}
The equation is now solved.
50x^{2}-95x+10098=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.
50x^{2}-95x+10098-10098=-10098
Subtract 10098 from both sides of the equation.
50x^{2}-95x=-10098
Subtracting 10098 from itself leaves 0.
\frac{50x^{2}-95x}{50}=-\frac{10098}{50}
Divide both sides by 50.
x^{2}+\left(-\frac{95}{50}\right)x=-\frac{10098}{50}
Dividing by 50 undoes the multiplication by 50.
x^{2}-\frac{19}{10}x=-\frac{10098}{50}
Reduce the fraction \frac{-95}{50} to lowest terms by extracting and canceling out 5.
x^{2}-\frac{19}{10}x=-\frac{5049}{25}
Reduce the fraction \frac{-10098}{50} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{19}{10}x+\left(-\frac{19}{20}\right)^{2}=-\frac{5049}{25}+\left(-\frac{19}{20}\right)^{2}
Divide -\frac{19}{10}, the coefficient of the x term, by 2 to get -\frac{19}{20}. Then add the square of -\frac{19}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{19}{10}x+\frac{361}{400}=-\frac{5049}{25}+\frac{361}{400}
Square -\frac{19}{20} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{19}{10}x+\frac{361}{400}=-\frac{80423}{400}
Add -\frac{5049}{25} to \frac{361}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{19}{20}\right)^{2}=-\frac{80423}{400}
Factor x^{2}-\frac{19}{10}x+\frac{361}{400}. 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{19}{20}\right)^{2}}=\sqrt{-\frac{80423}{400}}
Take the square root of both sides of the equation.
x-\frac{19}{20}=\frac{\sqrt{80423}i}{20} x-\frac{19}{20}=-\frac{\sqrt{80423}i}{20}
Simplify.
x=\frac{19+\sqrt{80423}i}{20} x=\frac{-\sqrt{80423}i+19}{20}
Add \frac{19}{20} to both sides of the equation.
x ^ 2 -\frac{19}{10}x +\frac{5049}{25} = 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 50
r + s = \frac{19}{10} rs = \frac{5049}{25}
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{19}{20} - u s = \frac{19}{20} + u
Two numbers r and s sum up to \frac{19}{10} exactly when the average of the two numbers is \frac{1}{2}*\frac{19}{10} = \frac{19}{20}. 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>
(\frac{19}{20} - u) (\frac{19}{20} + u) = \frac{5049}{25}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{5049}{25}
\frac{361}{400} - u^2 = \frac{5049}{25}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{5049}{25}-\frac{361}{400} = \frac{80423}{400}
Simplify the expression by subtracting \frac{361}{400} on both sides
u^2 = -\frac{80423}{400} u = \pm\sqrt{-\frac{80423}{400}} = \pm \frac{\sqrt{80423}}{20}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{19}{20} - \frac{\sqrt{80423}}{20}i = 0.950 - 14.179i s = \frac{19}{20} + \frac{\sqrt{80423}}{20}i = 0.950 + 14.179i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}