Solve for x (complex solution)
x=\frac{3+5\sqrt{23}i}{4}\approx 0.75+5.994789404i
x=\frac{-5\sqrt{23}i+3}{4}\approx 0.75-5.994789404i
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2x^{2}-3x+73=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(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\times 73}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -3 for b, and 73 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\times 73}}{2\times 2}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9-8\times 73}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-3\right)±\sqrt{9-584}}{2\times 2}
Multiply -8 times 73.
x=\frac{-\left(-3\right)±\sqrt{-575}}{2\times 2}
Add 9 to -584.
x=\frac{-\left(-3\right)±5\sqrt{23}i}{2\times 2}
Take the square root of -575.
x=\frac{3±5\sqrt{23}i}{2\times 2}
The opposite of -3 is 3.
x=\frac{3±5\sqrt{23}i}{4}
Multiply 2 times 2.
x=\frac{3+5\sqrt{23}i}{4}
Now solve the equation x=\frac{3±5\sqrt{23}i}{4} when ± is plus. Add 3 to 5i\sqrt{23}.
x=\frac{-5\sqrt{23}i+3}{4}
Now solve the equation x=\frac{3±5\sqrt{23}i}{4} when ± is minus. Subtract 5i\sqrt{23} from 3.
x=\frac{3+5\sqrt{23}i}{4} x=\frac{-5\sqrt{23}i+3}{4}
The equation is now solved.
2x^{2}-3x+73=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.
2x^{2}-3x+73-73=-73
Subtract 73 from both sides of the equation.
2x^{2}-3x=-73
Subtracting 73 from itself leaves 0.
\frac{2x^{2}-3x}{2}=-\frac{73}{2}
Divide both sides by 2.
x^{2}-\frac{3}{2}x=-\frac{73}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{73}{2}+\left(-\frac{3}{4}\right)^{2}
Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{73}{2}+\frac{9}{16}
Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{575}{16}
Add -\frac{73}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{3}{4}\right)^{2}=-\frac{575}{16}
Factor x^{2}-\frac{3}{2}x+\frac{9}{16}. 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{3}{4}\right)^{2}}=\sqrt{-\frac{575}{16}}
Take the square root of both sides of the equation.
x-\frac{3}{4}=\frac{5\sqrt{23}i}{4} x-\frac{3}{4}=-\frac{5\sqrt{23}i}{4}
Simplify.
x=\frac{3+5\sqrt{23}i}{4} x=\frac{-5\sqrt{23}i+3}{4}
Add \frac{3}{4} to both sides of the equation.
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}