Solve for x (complex solution)
x=\frac{-9+\sqrt{95}i}{22}\approx -0.409090909+0.443036107i
x=\frac{-\sqrt{95}i-9}{22}\approx -0.409090909-0.443036107i
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11x^{2}+9x+4=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{-9±\sqrt{9^{2}-4\times 11\times 4}}{2\times 11}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, 9 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-9±\sqrt{81-4\times 11\times 4}}{2\times 11}
Square 9.
x=\frac{-9±\sqrt{81-44\times 4}}{2\times 11}
Multiply -4 times 11.
x=\frac{-9±\sqrt{81-176}}{2\times 11}
Multiply -44 times 4.
x=\frac{-9±\sqrt{-95}}{2\times 11}
Add 81 to -176.
x=\frac{-9±\sqrt{95}i}{2\times 11}
Take the square root of -95.
x=\frac{-9±\sqrt{95}i}{22}
Multiply 2 times 11.
x=\frac{-9+\sqrt{95}i}{22}
Now solve the equation x=\frac{-9±\sqrt{95}i}{22} when ± is plus. Add -9 to i\sqrt{95}.
x=\frac{-\sqrt{95}i-9}{22}
Now solve the equation x=\frac{-9±\sqrt{95}i}{22} when ± is minus. Subtract i\sqrt{95} from -9.
x=\frac{-9+\sqrt{95}i}{22} x=\frac{-\sqrt{95}i-9}{22}
The equation is now solved.
11x^{2}+9x+4=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.
11x^{2}+9x+4-4=-4
Subtract 4 from both sides of the equation.
11x^{2}+9x=-4
Subtracting 4 from itself leaves 0.
\frac{11x^{2}+9x}{11}=-\frac{4}{11}
Divide both sides by 11.
x^{2}+\frac{9}{11}x=-\frac{4}{11}
Dividing by 11 undoes the multiplication by 11.
x^{2}+\frac{9}{11}x+\left(\frac{9}{22}\right)^{2}=-\frac{4}{11}+\left(\frac{9}{22}\right)^{2}
Divide \frac{9}{11}, the coefficient of the x term, by 2 to get \frac{9}{22}. Then add the square of \frac{9}{22} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{9}{11}x+\frac{81}{484}=-\frac{4}{11}+\frac{81}{484}
Square \frac{9}{22} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{9}{11}x+\frac{81}{484}=-\frac{95}{484}
Add -\frac{4}{11} to \frac{81}{484} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{9}{22}\right)^{2}=-\frac{95}{484}
Factor x^{2}+\frac{9}{11}x+\frac{81}{484}. 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{9}{22}\right)^{2}}=\sqrt{-\frac{95}{484}}
Take the square root of both sides of the equation.
x+\frac{9}{22}=\frac{\sqrt{95}i}{22} x+\frac{9}{22}=-\frac{\sqrt{95}i}{22}
Simplify.
x=\frac{-9+\sqrt{95}i}{22} x=\frac{-\sqrt{95}i-9}{22}
Subtract \frac{9}{22} from 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}