Solve for x (complex solution)
x=\frac{-\sqrt{51}i+3}{10}\approx 0.3-0.714142843i
x=\frac{3+\sqrt{51}i}{10}\approx 0.3+0.714142843i
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-5x^{2}+3x=3
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.
-5x^{2}+3x-3=3-3
Subtract 3 from both sides of the equation.
-5x^{2}+3x-3=0
Subtracting 3 from itself leaves 0.
x=\frac{-3±\sqrt{3^{2}-4\left(-5\right)\left(-3\right)}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 3 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4\left(-5\right)\left(-3\right)}}{2\left(-5\right)}
Square 3.
x=\frac{-3±\sqrt{9+20\left(-3\right)}}{2\left(-5\right)}
Multiply -4 times -5.
x=\frac{-3±\sqrt{9-60}}{2\left(-5\right)}
Multiply 20 times -3.
x=\frac{-3±\sqrt{-51}}{2\left(-5\right)}
Add 9 to -60.
x=\frac{-3±\sqrt{51}i}{2\left(-5\right)}
Take the square root of -51.
x=\frac{-3±\sqrt{51}i}{-10}
Multiply 2 times -5.
x=\frac{-3+\sqrt{51}i}{-10}
Now solve the equation x=\frac{-3±\sqrt{51}i}{-10} when ± is plus. Add -3 to i\sqrt{51}.
x=\frac{-\sqrt{51}i+3}{10}
Divide -3+i\sqrt{51} by -10.
x=\frac{-\sqrt{51}i-3}{-10}
Now solve the equation x=\frac{-3±\sqrt{51}i}{-10} when ± is minus. Subtract i\sqrt{51} from -3.
x=\frac{3+\sqrt{51}i}{10}
Divide -3-i\sqrt{51} by -10.
x=\frac{-\sqrt{51}i+3}{10} x=\frac{3+\sqrt{51}i}{10}
The equation is now solved.
-5x^{2}+3x=3
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.
\frac{-5x^{2}+3x}{-5}=\frac{3}{-5}
Divide both sides by -5.
x^{2}+\frac{3}{-5}x=\frac{3}{-5}
Dividing by -5 undoes the multiplication by -5.
x^{2}-\frac{3}{5}x=\frac{3}{-5}
Divide 3 by -5.
x^{2}-\frac{3}{5}x=-\frac{3}{5}
Divide 3 by -5.
x^{2}-\frac{3}{5}x+\left(-\frac{3}{10}\right)^{2}=-\frac{3}{5}+\left(-\frac{3}{10}\right)^{2}
Divide -\frac{3}{5}, the coefficient of the x term, by 2 to get -\frac{3}{10}. Then add the square of -\frac{3}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{3}{5}+\frac{9}{100}
Square -\frac{3}{10} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{51}{100}
Add -\frac{3}{5} to \frac{9}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{3}{10}\right)^{2}=-\frac{51}{100}
Factor x^{2}-\frac{3}{5}x+\frac{9}{100}. 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}{10}\right)^{2}}=\sqrt{-\frac{51}{100}}
Take the square root of both sides of the equation.
x-\frac{3}{10}=\frac{\sqrt{51}i}{10} x-\frac{3}{10}=-\frac{\sqrt{51}i}{10}
Simplify.
x=\frac{3+\sqrt{51}i}{10} x=\frac{-\sqrt{51}i+3}{10}
Add \frac{3}{10} to both sides of the equation.
Examples
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{ 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}