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3x^{2}+\frac{7}{8}x+\frac{1}{32}=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{-\frac{7}{8}±\sqrt{\left(\frac{7}{8}\right)^{2}-4\times 3\times \frac{1}{32}}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, \frac{7}{8} for b, and \frac{1}{32} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\frac{7}{8}±\sqrt{\frac{49}{64}-4\times 3\times \frac{1}{32}}}{2\times 3}
Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.
x=\frac{-\frac{7}{8}±\sqrt{\frac{49}{64}-12\times \frac{1}{32}}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\frac{7}{8}±\sqrt{\frac{49}{64}-\frac{3}{8}}}{2\times 3}
Multiply -12 times \frac{1}{32}.
x=\frac{-\frac{7}{8}±\sqrt{\frac{25}{64}}}{2\times 3}
Add \frac{49}{64} to -\frac{3}{8} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{-\frac{7}{8}±\frac{5}{8}}{2\times 3}
Take the square root of \frac{25}{64}.
x=\frac{-\frac{7}{8}±\frac{5}{8}}{6}
Multiply 2 times 3.
x=-\frac{\frac{1}{4}}{6}
Now solve the equation x=\frac{-\frac{7}{8}±\frac{5}{8}}{6} when ± is plus. Add -\frac{7}{8} to \frac{5}{8} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=-\frac{1}{24}
Divide -\frac{1}{4} by 6.
x=-\frac{\frac{3}{2}}{6}
Now solve the equation x=\frac{-\frac{7}{8}±\frac{5}{8}}{6} when ± is minus. Subtract \frac{5}{8} from -\frac{7}{8} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
x=-\frac{1}{4}
Divide -\frac{3}{2} by 6.
x=-\frac{1}{24} x=-\frac{1}{4}
The equation is now solved.
3x^{2}+\frac{7}{8}x+\frac{1}{32}=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.
3x^{2}+\frac{7}{8}x+\frac{1}{32}-\frac{1}{32}=-\frac{1}{32}
Subtract \frac{1}{32} from both sides of the equation.
3x^{2}+\frac{7}{8}x=-\frac{1}{32}
Subtracting \frac{1}{32} from itself leaves 0.
\frac{3x^{2}+\frac{7}{8}x}{3}=-\frac{\frac{1}{32}}{3}
Divide both sides by 3.
x^{2}+\frac{\frac{7}{8}}{3}x=-\frac{\frac{1}{32}}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}+\frac{7}{24}x=-\frac{\frac{1}{32}}{3}
Divide \frac{7}{8} by 3.
x^{2}+\frac{7}{24}x=-\frac{1}{96}
Divide -\frac{1}{32} by 3.
x^{2}+\frac{7}{24}x+\left(\frac{7}{48}\right)^{2}=-\frac{1}{96}+\left(\frac{7}{48}\right)^{2}
Divide \frac{7}{24}, the coefficient of the x term, by 2 to get \frac{7}{48}. Then add the square of \frac{7}{48} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\frac{7}{24}x+\frac{49}{2304}=-\frac{1}{96}+\frac{49}{2304}
Square \frac{7}{48} by squaring both the numerator and the denominator of the fraction.
x^{2}+\frac{7}{24}x+\frac{49}{2304}=\frac{25}{2304}
Add -\frac{1}{96} to \frac{49}{2304} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+\frac{7}{48}\right)^{2}=\frac{25}{2304}
Factor x^{2}+\frac{7}{24}x+\frac{49}{2304}. 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{7}{48}\right)^{2}}=\sqrt{\frac{25}{2304}}
Take the square root of both sides of the equation.
x+\frac{7}{48}=\frac{5}{48} x+\frac{7}{48}=-\frac{5}{48}
Simplify.
x=-\frac{1}{24} x=-\frac{1}{4}
Subtract \frac{7}{48} from both sides of the equation.