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$-16 \exponential{(x)}{2} - 4 x + 382 = 0 $
Solve for x
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-16x^{2}-4x+382=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(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-16\right)\times 382}}{2\left(-16\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, -4 for b, and 382 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-4\right)±\sqrt{16-4\left(-16\right)\times 382}}{2\left(-16\right)}
Square -4.
x=\frac{-\left(-4\right)±\sqrt{16+64\times 382}}{2\left(-16\right)}
Multiply -4 times -16.
x=\frac{-\left(-4\right)±\sqrt{16+24448}}{2\left(-16\right)}
Multiply 64 times 382.
x=\frac{-\left(-4\right)±\sqrt{24464}}{2\left(-16\right)}
Add 16 to 24448.
x=\frac{-\left(-4\right)±4\sqrt{1529}}{2\left(-16\right)}
Take the square root of 24464.
x=\frac{4±4\sqrt{1529}}{2\left(-16\right)}
The opposite of -4 is 4.
x=\frac{4±4\sqrt{1529}}{-32}
Multiply 2 times -16.
x=\frac{4\sqrt{1529}+4}{-32}
Now solve the equation x=\frac{4±4\sqrt{1529}}{-32} when ± is plus. Add 4 to 4\sqrt{1529}.
x=\frac{-\sqrt{1529}-1}{8}
Divide 4+4\sqrt{1529} by -32.
x=\frac{4-4\sqrt{1529}}{-32}
Now solve the equation x=\frac{4±4\sqrt{1529}}{-32} when ± is minus. Subtract 4\sqrt{1529} from 4.
x=\frac{\sqrt{1529}-1}{8}
Divide 4-4\sqrt{1529} by -32.
x=\frac{-\sqrt{1529}-1}{8} x=\frac{\sqrt{1529}-1}{8}
The equation is now solved.
-16x^{2}-4x+382=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.
-16x^{2}-4x+382-382=-382
Subtract 382 from both sides of the equation.
-16x^{2}-4x=-382
Subtracting 382 from itself leaves 0.
\frac{-16x^{2}-4x}{-16}=-\frac{382}{-16}
Divide both sides by -16.
x^{2}+\left(-\frac{4}{-16}\right)x=-\frac{382}{-16}
Dividing by -16 undoes the multiplication by -16.
x^{2}+\frac{1}{4}x=-\frac{382}{-16}
Reduce the fraction \frac{-4}{-16} to lowest terms by extracting and canceling out 4.
x^{2}+\frac{1}{4}x=\frac{191}{8}
Reduce the fraction \frac{-382}{-16} to lowest terms by extracting and canceling out 2.
x^{2}+\frac{1}{4}x+\left(\frac{1}{8}\right)^{2}=\frac{191}{8}+\left(\frac{1}{8}\right)^{2}
Divide \frac{1}{4}, the coefficient of the x term, by 2 to get \frac{1}{8}. Then add the square of \frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{191}{8}+\frac{1}{64}
Square \frac{1}{8} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{1529}{64}
Add \frac{191}{8} to \frac{1}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{1}{8}\right)^{2}=\frac{1529}{64}
Factor x^{2}+\frac{1}{4}x+\frac{1}{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{1}{8}\right)^{2}}=\sqrt{\frac{1529}{64}}
Take the square root of both sides of the equation.
x+\frac{1}{8}=\frac{\sqrt{1529}}{8} x+\frac{1}{8}=-\frac{\sqrt{1529}}{8}
Simplify.
x=\frac{\sqrt{1529}-1}{8} x=\frac{-\sqrt{1529}-1}{8}
Subtract \frac{1}{8} from both sides of the equation.