Solve for x (complex solution)
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4x^{2}-11x+30=16
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.
4x^{2}-11x+30-16=16-16
Subtract 16 from both sides of the equation.
4x^{2}-11x+30-16=0
Subtracting 16 from itself leaves 0.
4x^{2}-11x+14=0
Subtract 16 from 30.
x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 4\times 14}}{2\times 4}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -11 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-11\right)±\sqrt{121-4\times 4\times 14}}{2\times 4}
Square -11.
x=\frac{-\left(-11\right)±\sqrt{121-16\times 14}}{2\times 4}
Multiply -4 times 4.
x=\frac{-\left(-11\right)±\sqrt{121-224}}{2\times 4}
Multiply -16 times 14.
x=\frac{-\left(-11\right)±\sqrt{-103}}{2\times 4}
Add 121 to -224.
x=\frac{-\left(-11\right)±\sqrt{103}i}{2\times 4}
Take the square root of -103.
x=\frac{11±\sqrt{103}i}{2\times 4}
The opposite of -11 is 11.
x=\frac{11±\sqrt{103}i}{8}
Multiply 2 times 4.
x=\frac{11+\sqrt{103}i}{8}
Now solve the equation x=\frac{11±\sqrt{103}i}{8} when ± is plus. Add 11 to i\sqrt{103}.
x=\frac{-\sqrt{103}i+11}{8}
Now solve the equation x=\frac{11±\sqrt{103}i}{8} when ± is minus. Subtract i\sqrt{103} from 11.
x=\frac{11+\sqrt{103}i}{8} x=\frac{-\sqrt{103}i+11}{8}
The equation is now solved.
4x^{2}-11x+30=16
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.
4x^{2}-11x+30-30=16-30
Subtract 30 from both sides of the equation.
4x^{2}-11x=16-30
Subtracting 30 from itself leaves 0.
4x^{2}-11x=-14
Subtract 30 from 16.
\frac{4x^{2}-11x}{4}=\frac{-14}{4}
Divide both sides by 4.
x^{2}+\frac{-11}{4}x=\frac{-14}{4}
Dividing by 4 undoes the multiplication by 4.
x^{2}-\frac{11}{4}x=\frac{-14}{4}
Divide -11 by 4.
x^{2}-\frac{11}{4}x=-\frac{7}{2}
Reduce the fraction \frac{-14}{4} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=-\frac{7}{2}+\left(-\frac{11}{8}\right)^{2}
Divide -\frac{11}{4}, the coefficient of the x term, by 2 to get -\frac{11}{8}. Then add the square of -\frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{7}{2}+\frac{121}{64}
Square -\frac{11}{8} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{103}{64}
Add -\frac{7}{2} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{11}{8}\right)^{2}=-\frac{103}{64}
Factor x^{2}-\frac{11}{4}x+\frac{121}{64}. 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{11}{8}\right)^{2}}=\sqrt{-\frac{103}{64}}
Take the square root of both sides of the equation.
x-\frac{11}{8}=\frac{\sqrt{103}i}{8} x-\frac{11}{8}=-\frac{\sqrt{103}i}{8}
Simplify.
x=\frac{11+\sqrt{103}i}{8} x=\frac{-\sqrt{103}i+11}{8}
Add \frac{11}{8} to both sides of the equation.